REPRESENTATION THEORY AND THE FOURIER-

STIELTJES TRANSFORM FOR COMPACT GROUPS

A Thesis

Submitted to the Graduate Faculty

of the

North Dakota State University of Agriculture and Applied Science

By

William Riley Casper

In Partial Fulfillment of the Requirements

for the Degree of MASTER OF SCIENCE

Major Department: Mathematics

December 2009

Fargo, North Dakota

ABSTRACT

Casper, William Riley, M.S., Department of Mathematics, College of Science and Mathematics, North Dakota State University, December 2009. . Major Professor: Dr. James H. Olsen.

In this paper, we set up the basic definitions of the theory representations of locally compact topological groups. As necessary background material we define the normed linear space M(G) of all complex measures on G and show that there is a linear isometry between this space and the dual of the space C_{0}(G) of all continuous functions on G going to zero at infinity. Using this relationship, we then define the convolution and adjoint of complex measures on G. We then discuss the theory of representations of a locally compact group G and in particular explain a means of “extending” representations of G to representations of M(G) and “restricting” representations of M(G) to representations of G. We use this to obtain the Gel’fand-Raĭkov theorem which shows that the collection of all irreducible, continuous unitary representations of a locally compact group “separates points”. This is then used to prove the Peter-Weyl theorem which establishes that the collection of coordinate of a compact group forms an orthonormal basis. The paper culminates with the definition of the Fourier-Stieltjes transform, as defined in [1].

ACKNOWLEDGMENTS

First of all, I would like to thank Dr. James H. Olsen and Dr. Friedrich Littmann for exhibiting incredible persistence and patience in their long-suffering attendance of our weekly seminar on abstract harmonic analysis. They deserve thanks in more ways than can be mentioned here. Specifically, I would like to thank my adviser, Dr. Olsen, for providing me with many interesting papers to read, providing examples of applications of the theory, and helping me remember on several occasions which way is up. I would like to thank my friends and family who managed to put up with my complaining, to listen to various explanations of the material, and to help in various ways to make this hefty stack of paper exist. I would like to thank my committee for endeavoring to wade through this heap of material. Lastly, I would like to thank Chuck Norris for allowing me to live long enough to finish this paper.

TABLE OF CONTENTS

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
CHAPTER 1. COMPLEX MEASURES ON G . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1. Introduction and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2. Properties of Complex Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3. An Isometry Between M(X) and C_{0}(X)^{∗} . . . . . . . . . . . . . . . . . . . . . . 10
1.4. Decomposition Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5. Product Measures and Fubini’s Theorem . . . . . . . . . . . . . . . . . . . . . . 21
1.6. The ∗-Algebra M(G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.7. Decomposition of M(G) and Adjoint Operators . . . . . . . . . . . . . . . . 32
CHAPTER 2. REPRESENTATIONS OF LOCALLY COMPACT GROUPS 42
2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.2. Basic Definitions and Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.3. Extensions of Representations to M(G) . . . . . . . . . . . . . . . . . . . . . . . 45
2.4. Continuity of Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.5. Restriction of Representations of M(G) . . . . . . . . . . . . . . . . . . . . . . . 53
2.6. The Regular Representation of M(G) and Gel’fand-Raĭkov . . . . . . . 60
CHAPTER 3. UNITARY REPRESENTATIONS OF COMPACT GROUPS 68
3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.2. Basic Definitions and Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.3. Trigonometric Polynomials and Stone-Weierstrass . . . . . . . . . . . . . . . 71

3.4. Orthogonality Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.5. The Fourier-Stieltjes Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

CHAPTER 1. COMPLEX MEASURES ON G

1.1. Introduction and Notation

This chapter is an introduction to the theory of complex measures generated by linear functionals of the Banach space C_{0}(X) of all continuous, complex-valued functions on a locally compact Hausdorff space X vanishing at infinity. Throughout this chapter, we assume that X is a locally compact Hausdorff space and that G is a locally compact T_{0} topological group. Thus results obtained for X will be able to be applied immediately to G^{1}. We will let C_{0}(X) denote the collection of all complex-valued functions on X that are arbitrarily small outside a compact set, C_{00}(X) denote the collection of all complex-valued functions with support contained in a compact set, and 𝔐 the collection of all lower semicontinuous functions on X. Given any linear space of complex-valued functions S, we denote by S^{∗} the collection of all bounded linear functionals (the dual space) on S. Additionally, we use S^{r} and S^{+} to denote the collections of all real and positive functions. For f, g ∈ S^{r}, we define the functions min(f, g) and max(f, g) (which may or may not be in S) by

min(f, g)(x) = \frac{1}{2}(f(x) + g(x) − |f(x) − g(x)|) = \{ \begin{matrix}f(x), & f(x) ⩽ g(x); \\ g(x), & otherwise \end{matrix} , (1)

and

max(f, g)(x) = \frac{1}{2}(f(x) + g(x) + |f(x) − g(x)|) = \{ \begin{matrix}f(x), & f(x) ⩾ g(x); \\ g(x), & otherwise \end{matrix} . (2)

One should be careful not to confuse this later with the definitions of max(T_{1}, T_{2}) and min(T_{1}, T_{2}) for linear functionals. We will us Re(f) and Im(f) to denote the real and imaginary components of a function f ∈ S

1All T_{0} locally compact topological groups are Hausdorff. See [2] for details.

Definition 1. A topological group is a group G with a topology under which the group operations

(i) G × G → G : (x, y) ↦ xy

(ii) G → G : x ↦ x^{−1}

are continuous. A topological group G is called locally compact if it is locally compact as a topological space. That is, if every member x ∈ G has an open neighborhood contained in a compact set.

We will accept without proof various properties of locally compact topological groups, including the existence of a (left or right) Haar measure. Unless otherwise stated, λ will always denote a left Haar measure on G. If G is compact, we will choose λ to be the unique Haar measure such that λ(G) = 1. The symbols τ : G × G → G and θ : G → G defined by (x, y) ↦ xy and x ↦ x^{−1} will always be used to denote the continuous mappings of multiplication and inversion on G, unless stated otherwise.

In this chapter we will establish a method for constructing a complex measure from a linear functional T ∈ C_{0}^{∗}(X) and obtain a linear isomorphism Φ : C_{0}^{∗}(X) → M(X). The linear space M(X) can be given a norm defined via the total variation |μ| of a measure μ ∈ M(X). In particular, the norm is defined by ∥μ∥ = |μ|(X). Under this definition, the mapping Φ is an isometry. During the second half of the chapter, we consider the specific case where X = G. We will define two additional natural operations for the space C_{0}^{∗}(G): convolution and adjoint. Via the transformation Φ, these operations will be extended to operations on M(G). In particular, defining these operations will allow us to consider M(G) as a ∗-algebra, a fact that will play a central role in the construction of continuous, unitary representations of locally compact groups later on.

1.2. Properties of Complex Measures

In this section, we establish some results for complex measures necessary for the rest of the chapter, as well as for some results in future chapters. In particular, we will define the total variation of a complex measure and use it to define a norm on M(X).

By a complex Borel measure on X, we will always be considering finite measures: given a complex Borel measure μ, the series ∑_{i=1}^{∞} μ(U_{i}) converges absolutely for any sequence {U_{i}}_{i=1}^{∞} of mutually disjoint Borel subsets of X. This is formalized in the next definition.

Definition 2. A complex measure on the Borel σ-algebra 𝔅(X) is a set function μ satisfying

(i) μ(∅) = 0;

(ii) given any sequence of mutually disjoint subsets {U_{i}}_{i=1}^{∞} ⊂ 𝔅(X), the series ∑_{i=1}^{∞} μ(U_{i}) converges absolutely to μ(⋃_{i=1}^{∞} U_{i}).

We will denote the collection of all complex Borel measures on X as M(X). For μ, ν ∈ M(X) and α ∈ ℂ, we define αμ and μ + ν by

(αμ)(U) = αμ(U);
(μ + ν)(U) = μ(U) + ν(U),

for all U ∈ 𝔅(X). Under this definition, αμ and μ+ν are also complex Borel measures on X, and M(X) is a complex linear space.

Definition 3. Given a Borel set U ∈ 𝔅(X), a finite Borel partition P of U is a finite collection of Borel sets P ⊂ 𝔅(X) that are mutually disjoint. The total variation of a complex measure μ is the finite, real-valued Borel measure defined by

|μ|(U) = sup {∑_{U∈P} |μ(U_{i})| : P is a finite Borel partition of U}  (3)

for all U ∈ 𝔅(X).

In the above, the choice of using finite partitions is somewhat arbitrary; infinite partitions would construct the same measure. Also, for our purposes, we will not be needing any other types of partitions, so we will refer to finite Borel partitions simply as “partitions”. It is definitely not clear from the definition above that |μ| is a measure, let alone finite. Convincing us of this is the burden of the next theorem.

Theorem 4. The positive set function μ defined by Equation (3) is a finite Borel measure on X.

Proof. It is clear from the definition that |μ|(∅) = 0. Moreover, let {U_{i}}_{i=1}^{∞} ⊂ 𝔅(X) be a sequence of mutually disjoint sets and define U = ⋃_{i=1}^{∞} U_{i}. Then given a partition P of U, each collection {V ∩ U_{j}}_{V ∈P} is a partition of U_{j} and therefore

∑_{V ∈P} |μ(V)| = ∑_{V ∈P}^{n} |∑_{j=1}^{∞} μ(U_{j} ∩ V)| ⩽ ∑_{j=1}^{∞} ∑_{V ∈P} |μ(U_{j} ∩ V)| ⩽ ∑_{j=1}^{∞} |μ|(U_{j}).

Since the partition of U was chosen arbitrarily, we may conclude

|μ|(U) ⩽ ∑_{j=1}^{∞} |μ|(U_{j}).

To prove the converse inequality, let ϵ > 0. Then for each integer j ⩾ 1, there exists a partition P_{j} of U_{j} such that

|μ|(U_{j}) ⩽ ∑_{V ∈P_{j}} |μ(V)| + ϵ2^{−j}.

Now for any integer k ⩾ 1, define V_{k} = ⋃_{j=k}^{∞} U_{j}. We have that for all integers k ⩾ 1, the collection {V : 1 ⩽ j ⩽ k, V ∈ P_{j}} ∪ {V_{k+1}} forms a partition of U so that

∑_{j=1}^{k} |μ|(U_{j}) ⩽ ∑_{j=1}^{k} ( ∑_{V ∈P_{j}} |μ(V)| + ϵ2^{−j}) ⩽ |μ(V_{k+1})| + ∑_{j=1}^{k} ( ∑_{V ∈P_{j}} |μ(V)| + ϵ2^{−j})

⩽ |μ|(U) + ϵ ∑_{j=1}^{k} 2^{−j} < |μ|(U) + ϵ.

Taking the limit as k → ∞ and then as ϵ → 0 provides us with the reverse inequality, and we conclude

|μ|(U) = ∑_{j=1}^{∞} |μ|(U_{j}).

Hence |μ| is a measure. If |μ| is not finite, then |μ|(U) = ∞ for some U ∈ 𝔅(X). Thus for all integers j ⩾ 1, there exists a partition P_{j} of U such that ∑_{V ∈P_{j}} |μ(V)| > j. Taking the limit as j → ∞, we obtain a countable partition P of U such that ∑_{V ∈P} |μ(V)| = ∞. This contradicts the assumption that μ is a complex measure.

We use the total variation |μ| of a complex measure μ to determine which functions are integrable with respect to μ. In particular, we have the following definition.

Definition 5. If ν is a positive measure, then a ν-measureable function f is said to be ν-integrable if ∫ |f|dν < ∞, and we write f ∈ L_{1}(ν). If μ is a complex measure, f is said to be μ-integrable if f ∈ L_{1}(|μ|), and we write f ∈ L_{1}(μ). That is, L_{1}(μ) = L_{1}(|μ|) by definition.

If f ∈ L_{1}(|μ|) and ν is the measure defined by ν(U) = ∫_{U} fdμ, we use the usual notation dν = fdμ. Before exploring the properties of the total variation, it is useful to recall an important result from measure theory.

Definition 6. Let μ and ν be measures on the same measure space X. Then μ is absolutely continuous with respect to ν, denoted (μ ≪ ν) if μ(U) = 0 for all sets U ⊂ X of ν-measure 0. Definition 7. Let μ and ν be measures on the same measure space X. Then μ and ν are said to be mutually singular, denoted μ ⊥ ν, if there exist measureable sets U_{μ}, U_{ν} ⊂ X satisfying U_{μ} ∩ U_{ν} = ∅, U_{μ} ∪ U_{ν} = X, μ(U) = 0 for all measureable U ⊂ U_{ν}, and ν(U) = 0 for all measureable U ⊂ U_{μ}. That is, the measures μ and ν ’live on different sets’. Theorem 8 (Lebesgue-Radon-Nikodym Theorem). If μ is a complex measure and ν is a σ-finite positive measure on X, there exists a complex measure λ and a function f ∈ L_{1}(ν) such that λ and μ are mutually singular and dν = dλ + fdμ. Moreover, the function f and the measure λ are unique up to sets of μ-measure 0. Proof. See, for example, [3] pp. 90-93.

In the case that μ ≪ ν, the above theorem guarantees the existence of a measure f ∈ L_{1}(ν) such that dμ = fdν, and we use the notation f = dμ/dν. It is clear from the definition of |μ| that μ ≪ |μ|, so as a particular case, we are always guaranteed the existence of dμ/d|μ| ∈ L_{1}(|μ|). Using this tool, we are prepared for an investigation of the properties of |μ|. The first main property we establish is the existence of a norm. To do so, we first establish some preliminary results. Lemma 9. Let μ be a complex measure and ν a σ-finite positive measure with f ∈ L_{1}(ν) satisfying dμ = fdν. Then d|μ| = |f|dν. Proof. Let μ, ν and f satisfy the assumptions of the theorem, and let U ∈ 𝔅(X). Given any partition P of U, we have that

∑_{V ∈P} |μ(V)| = ∑_{V ∈P} |∫_{V} fdν| ⩽ ∑_{V ∈P} ∫_{V} |f|dν = ∫_{U} |f|dν.

Since the choice of partition was arbitrary, we may conclude that |μ|(U) ⩽ ∫_{U} |f|dν.

To prove the converse, suppose that U is compact. Then f may be approximated on U by simple functions^{2}. In particular, for all ϵ > 0, there exists a partition P of U and a collection of complex numbers {α_{V} : V ∈ P } such that |f − ∑_{V ∈P} α_{V} 1_{V} | < ϵ/2ν(U) on U. It follows that

∫_{U} |f|dν ⩽ ∑_{V ∈P} |α_{V} |ν(V) + ϵ/2 = |∑_{V ∈P} α_{V} ν(V)| + ϵ/2
⩽ |∫_{U} fdν| + ϵ = |μ(U)| + ϵ < |μ|(U) + ϵ.

Taking the limit as ϵ → 0, we find that ∫_{U} |f|dν ⩽ |μ|(U). Combining this with our previous inequality proves d|μ| = |f|dν for all compact sets. Since both |μ| and ν are σ-finite, this proves our lemma.

Corollary 10. Let μ ∈ M(X). Then |\frac{dμ}{d|μ|}(x)| = 1 for |μ|-a.e. x ∈ X.

Proof. As a direct result of the definition of total variation, the total variation of a finite positive measure is the measure itself. Since dμ = \frac{dμ}{d|μ|}d|μ|, Lemma (9) tells us

d|μ| = |\frac{dμ}{d|μ|}| d|μ|.

The statement of the corollary follows immediately.

Corollary 11. Let μ, ν ∈ M(X), and f ∈ L_{1}(|ν|) with dμ = fdν. Then d|μ| = |f|d|ν|.

Proof. Note that dμ = f \frac{dν}{d|ν|}d|ν|. Lemma (9) and the previous corollary tell us

d|μ| = |f \frac{dν}{d|ν|}| d|ν| = |f|d|ν|.

2This is a standard fact when f is real and positive. If f is complex, it can be accomplished by decomposing f into a linear combination of positive, real-valued functions.

Corollary 12. Let μ ∈ M(G) and f ∈ L_{1}(|μ|). Then

|∫_{X} fdμ| ⩽ ∫ |f|d|μ|.

Proof. Let dν = fdμ. Then ν ∈ M(X) and d|ν| = |f|d|μ|. It follows that

|∫_{X} fdμ| = |ν(X)| ⩽ |ν|(X) = ∫_{X} |f|d|μ|.

Theorem 13. Let μ_{1}, μ_{2} ∈ M(X). Then |μ_{1} + μ_{2}| ⩽ |μ_{1}| + |μ_{2}| and M(X) is a normed linear space with norm defined by

∥μ∥ = |μ|(X).  (4)

Proof. Define ν = |μ_{1}| + |μ_{2}|. Then μ_{i} ≪ ν and there exist functions f_{i} ∈ L_{1}(ν) such that dμ_{i} = f_{i}dν for i = 1, 2. By the previous lemma, d|μ_{i}| = |f_{i}|dν. Thus for any U ∈ 𝔅(X), we have that

|μ_{1}(U) + μ_{2}(U)| = |∫_{U} f_{1} + f_{2}dν| ⩽ ∫_{U} |f_{1}|dν + ∫_{U} |f_{2}|dν = |μ_{1}|(U) + |μ_{2}|(U).

In particular, this shows that ∥μ_{1} + μ_{2}∥ = |μ_{1} + μ_{2}|(X) ⩽ |μ_{1}|(X) + |μ_{2}|(X) = ∥μ_{1}∥ + ∥μ_{2}∥. Moreover, dμ_{1} = f_{1}dν, so that d(αμ_{1}) = αdμ_{1} = αf_{1}dν. It follows from the previous lemma that d|αμ_{1}| = |α||f_{1}|dν, and therefore ∥αμ_{1}∥ = |α|∥μ_{1}∥. Since the choice of μ_{1}, μ_{2} ∈ M(X) and α ∈ ℂ was arbitrary, all that is left to show is that ∥μ∥ ⩾ 0, with equality if and only if μ is the zero measure.

If μ ∈ M(X) is the zero measure, it follows from the definition that ∥μ∥ = 0.

If μ ≠ 0, then there exists a set U ∈ 𝔅(X) such that μ(U) ≠ 0. Therefore ∥μ∥ = |μ|(X) ⩾ |μ(U)| + |μ(X\U)| ⩾ |μ(U)| > 0. This proves our theorem.

The total variation of a measure μ may also be expressed in terms of functions in C_{0}(X). This is a very important fact, as it will be the key in relating the norm on M(X) to the norm on C_{0}^{∗}(X).

Theorem 14. Let μ ∈ M(X). Then for all U ∈ 𝔅(X) we have that

|μ|(U) = sup {|∫_{U} fdμ| : f ∈ C_{0}(X), |f| ⩽ 1} .  (5)

More generally, given any real, positive function f ∈ L_{1}(|μ|), we have that

∫_{X} fd|μ| = sup {|∫_{X} gdμ| : g ∈ C_{0}(X), |g| ⩽ f} .  (6)

Proof. Let U ∈ 𝔅(X) and define c = sup {|∫_{U} fdμ| : f ∈ C_{0}(X), |f| ⩽ 1}. Let ϵ > 0 and fix a partition P of U. For every V ∈ P, there exists a compact set V_{ϵ} ⊂ V such that μ(V △V_{ϵ}) < ϵ/n, where n is the number of members of P. Define φ = ∑_{i=1}^{n} sign(μ(V_{ϵ}))1_{V_{ϵ}} ∈ C_{0}(X), and let f ∈ C_{0}(X) satisfy |f − φ| < ϵ/|μ|(U) and |f| ⩽ 1. Then

∑_{V ∈P} |μ(V)| − ϵ < ∑_{V ∈P} |μ(V_{ϵ})| = ∫_{U} φdμ ⩽ ∫_{U} |f − φ|d|μ| + |∫_{U} fdμ| ⩽ c + ϵ.

Taking the limit as ϵ → 0, and noting that the partition P was arbitrary, we conclude |μ|(V) ⩽ c.

Conversely, let f ∈ C_{0}(X) with |f| ⩽ 1. Then | ∫_{U} fdμ| ⩽ ∫_{U} |f|d|μ| ⩽ |μ|(U). This proves the identity in Equation (5).

Let f ∈ C_{00}(X) be a real, positive function and define a complex measure ν by dν = fd|μ|. Suppose that h ∈ C_{0}(X) with |h| ⩽ f. Then define a function g by

g(x) = \{ \begin{matrix}h(x)/f(x), & for f(x) ≠ 0; \\ 0, & for f(x) = 0. \end{matrix}

Then g ∈ C_{00}(X), |g| ⩽ 1 and fg = h for |μ|-a.e. x ∈ X. Conversely, if g ∈ C_{0}(X) with |g| ⩽ 1, then fg ∈ C_{00}(X) and |fg| ⩽ f. It follows that

∫_{X} fd|μ| = sup {|∫_{X} gdν| : g ∈ C_{0}(X), |g| ⩽ 1}
= sup {|∫_{X} hdμ| : h ∈ C_{0}(X), |h| ⩽ f} .

This proves the identity in Equation (6) for real, positive f ∈ C_{00}(X). The identity for all real, positive f ∈ L_{1}(μ) follows immediately from the monotone convergence theorem by approximating f with functions in C_{00}.

1.3. An Isometry Between M(X) and C_{0}(X)^{∗}

For every μ ∈ M(X), there exists a linear functional T_{μ} ∈ C_{0}(X)^{∗} satisfying

⟨f, T_{μ}⟩ = ∫_{X} fdμ.  (7)

The primary content of this section is that every linear functional in C_{0}(X)^{∗} can be described in this way, and in fact that the correspondence Φ : M(X) → C_{0}(X)^{∗} defined by Φ : μ ↦ T_{μ} is an isometry.

We begin by proving that the mapping Φ is a well-defined norm-preserving monomorphism of Φ onto M(X). Theorem 15. Let μ ∈ M(X). Then T_{μ} defined by Equation (7) is a bounded linear functional in C_{0}(X) and ∥T_{μ}∥ = ∥μ∥. Moreover, the mapping Φ : μ ↦ T_{μ} is a norm-preserving monomorphism of M(X) into C_{0}^{∗}(X).

Proof. From the properties of integration, T_{μ} is a linear functional. From Equation (5)

∥μ∥ = sup {|∫_{X} fdμ| : f ∈ C_{0}(X), |f| ⩽ 1}
= {|⟨f, T_{μ}⟩| : f ∈ C_{0}(X), |f| ⩽ 1} = ∥T_{μ}∥,

and therefore Φ preserves norms. If μ, ν ∈ M(X) and α ∈ ℂ, then

⟨f, T_{μ+ν}⟩ = ∫_{X} fd(μ + ν) = ∫_{X} fdμ + ∫_{X} fdν = ⟨f, T_{μ}⟩ + ⟨f, T_{ν}⟩,

for all f ∈ C_{0}(X). Thus T_{μ+ν} = T_{μ} + T_{ν}, and similarly T_{αμ} = αT_{μ}. Thus Φ is a homomorphism. To show that it is a monomorphism, note that if T_{μ} = 0, then for any U ∈ 𝔅(X), ⟨1_{U}, T_{μ}⟩ = μ(U) = 0. It follows that μ = 0. This proves our theorem.

To show that Φ is an isometry, all that is left to show is that Φ is onto. Equivalently, this is saying that every linear functional T ∈ C_{0}^{∗}(X) is of the form Equation (7) with T = T_{μ} for some measure μ ∈ M(X). This is a standard fact in the case that T is a real, positive linear functional. The general case requires some machinery that will be used to express any linear functional T ∈ C_{0}^{∗}(X) as a linear combination of real, positive linear functionals. The key to this is the “linear modulus” of a linear operator.

In order to define the linear modulus, we first note that if B is a Banach space of complex-valued functions such that Re(f) and Im(f) are in B for every f ∈ B, then the value of a linear functional T on B is completely determined by its value on positive, real-valued functions. In particular,

T (f) = T (max(Re(f), 0)) − T (− min(Re(f), 0))
+ i(T (max(Im(f), 0)) − T (− min(Im(f), 0)))

for every f ∈ B. Thus, if we define a linear functional T on the subset of all realvalued functions in B it extends uniquely to a linear functional on B.

Definition 16. Let T be a bounded linear functional on a Banach space B of complexvalued functions such that Re(f) and Im(f) are in B for every f ∈ B. The linear modulus |T | of T is the functional on B defined by

|T |(f) = sup{|T (g)| : g ∈ B, |g| ⩽ |f|}  (8)

for all positive, real-valued functions f ∈ B.

This definition looks very similar to the definition of the total variation of a signed measure, and for good reason. In fact, we will be able to show that Φ : |μ| ↦ |T_{μ}|. However, before we do that, we should convince ourselves that |T | is in fact a bounded linear functional on B. This is the task of the next theorem.

Theorem 17. Let B be a Banach space of complex-valued functions such that Re(f) and Im(f) are in B for every f ∈ B, and let T be a bounded linear functional on B. Then |T | as defined by Equation (8) is a bounded, positive linear functional on B with |T |(f) ⩽ |T |(g) for real, positive functions f, g with f ⩽ g and ∥T ∥ = ∥|T |∥.

Proof. Suppose that f_{1}, f_{2} ∈ B are real-valued, positive functions. Then for any h ∈ B with |h| ⩽ f_{1} + f_{2} we have that the functions h_{1} and h_{2} defined by h_{2} = min(|h|, f_{2})sign(h) and h_{1} = (|h| − h_{2})sign(h) satisfy |h_{i}| < f_{i} and h_{1} + h_{2} = h. Thus

|T (h)| = |T (h_{1} + h_{2})| ⩽ |T (h_{1})| + |T (h_{2})| ⩽ |T |(f_{1}) + |T |(f_{2}).

Since this is true for any h ∈ B with |h| ⩽ f_{1}+f_{2}, we may conclude that |T |(f_{1}+f_{2}) ⩽ |T |(f_{1}) + |T |(f_{2}). Moreover, for all ϵ > 0, there exists h_{i} ∈ B such that |h_{i}| ⩽ f_{i} and |T |(f_{i}) ⩽ |T (h_{i})| + ϵ/2, for i = 1, 2. Define λ_{i} ∈ 𝕋 such that |T (h_{i})| = λ_{i}T (h_{i}). Then |λ_{1}h_{1} + λ_{2}h_{2}| ⩽ |h_{1}| + |h_{2}| ⩽ f_{1} + f_{2} and

|T |(f_{1}) + |T |(f_{2}) − ϵ < |T (h_{1})| + |T (h_{2})| = ||T (h_{1})| + |T (h_{2})||
= |λ_{1}T (h_{1}) + λ_{2}T (h_{2})| ⩽ |T |(f_{1} + f_{2}).

Thus |T |(f_{1})+|T |(f_{2}) ⩽ |T |(f_{1} +f_{2}). Combining this with the previous inequality, we conclude that |T |(f_{1}) + |T |(f_{2}) = |T |(f_{1} + f_{2}). Moreover, for any real number α > 0,

|T |(αf_{1}) = sup{|T (h_{1})| : |h_{1}| ⩽ αf_{1}} = sup{|T (αh_{1})| : |h_{1}| ⩽ f_{1}}
= sup{α|T (h_{1})| : |h_{1}| ⩽ f_{1}} = α|T |(f_{1}).

Hence |T | is a positive linear functional on the set of all real, positive functions in B. It follows that T is a positive linear functional on B. Moreover, the definition of T tells us that T (f_{1}) ⩽ T (f_{2}) for f_{1} ⩽ f_{2}.

The last thing that we must show is that ∥T ∥ = ∥|T |∥. We note that

∥|T |∥ = sup{||T |(f)| : f ∈ B, |f| ⩽ 1}

= sup{| sup{|T (g)| : g ∈ B, |g| ⩽ f}| : f ∈ B, |f| ⩽ 1}

= | sup{|T (g)| : g ∈ B, |g| ⩽ 1}| = |(∥T ∥)| = ∥T ∥.

This proves our theorem.

Using the linear modulus, we may construct “maximum” and “minimum” linear functionals from two arbitrary linear functionals. The notion of and notation for this are both provided in the next definition.

Definition 18. Let T_{1}, T_{2} be linear functionals on a Banach space B of complexvalued functions such that Re(f) and Im(f) are in B for every f ∈ B. We define the minimum min(T_{1}, T_{2}) and the maximum max(T_{1}, T_{2}) of two linear operators T_{1} and T_{2}, by

max(T_{1}, T_{2})(f) = \frac{1}{2}(T_{1}(f) + T_{2}(f) + |T_{1} + T_{2}|(f))  (9)
min(T_{1}, T_{2})(f) = \frac{1}{2}(T_{1}(f) + T_{2}(f) − |T_{1} + T_{2}|(f)).  (10)

It is clear from the definition max(T_{1}, T_{2}) and min(T_{1}, T_{2}) are linear functionals. In addition, we call two linear functionals T_{1} and T_{2} mutually singular if min(T_{1}, T_{2}) = 0.

The maximum and minimum defined above provides a partial ordering on the collection of all linear functionals on B. In particular, we say that T_{1} ⩽ T_{2} if min(T_{1}, T_{2}) = T_{1} or equivalently max(T_{1}, T_{2}) = T_{2}. The next theorem outlines some important properties of these linear functionals.

Theorem 19. Let T_{1}, T_{2} be linear functionals on a Banach space B of complex-valued functions. Then max(T_{1}, T_{2}) = max(T_{2}, T_{1}), min(T_{1}, T_{2}) = min(T_{2}, T_{1}), max(T_{1}, T_{2}) = max(T_{1} − T_{2}, 0) + T_{2}, and min(T_{1}, T_{2}) = min(T_{1} − T_{2}, 0) + T_{2}. Also min(T_{1}, T_{2}) = T_{1} − max(T_{1} − T_{2}, 0).

Proof. The properties max(T_{1}, T_{2}) = max(T_{2}, T_{1}) and min(T_{1}, T_{2}) = min(T_{2}, T_{1}) follow immediately from the definitions. Moreover

min(T_{1} − T_{2}, 0) + T_{2} = \frac{1}{2}(T_{1} − T_{2} + 0 − |T_{1} − T_{2} + 0|) + T_{2}

= \frac{1}{2}(T_{1} + T_{2} − |T_{1} − T_{2}|) = min(T_{1}, T_{2}).

The proof of max(T_{1}, T_{2}) = max(T_{1} − T_{2}, 0) + T_{2} is similar. Lastly, we have that

T_{1} − max(T_{1} − T_{2}, 0) = T_{1} − max(T_{1}, T_{2}) + T_{2} = (T_{1} + T_{2} − |T_{1} − T_{2}|) = min(T_{1}, T_{2}).

The machinery just defined may be used to rip apart a linear functional T into a linear combination of positive linear functionals that is unique in a certain sense.

Theorem 20. Let T ∈ C_{0}^{∗}(X). Then T may be decomposed into a direct sum T = T_{1} −T_{2} +i(T_{3} −T_{4}), where T_{i} is a positive bounded linear functional for 1 ⩽ i ⩽ 4 and the decomposition is unique in the sense that T_{1} and T_{2} are mutually singular and T_{3} and T_{4} are mutually singular. If T = T′_{1} − T′_{2} + i(T′_{3} − T′_{4}) for some positive bounded linear functionals T′_{i} and T′_{1} and T′_{2} are mutually singular, and T_{3} and T_{4} are mutually singular, then T_{i} = T′_{i} for 1 ⩽ i ⩽ 4.

Proof. Define linear functionals T_{R} and T_{I} by T_{R}(f) = Re(T (f)) and T_{I}(f) = Im(T (f)) for all real-valued, positive f ∈ C_{0}(X). Since T is bounded, so too are T_{R} and T_{I}. Define T_{1} = max(T_{R}, 0), T_{2} = T_{1} − T_{R}, T_{3} = max(T_{I}, 0) and T_{4} = T_{3} − T_{I}. Then T_{i} is a bounded linear functional for 1 ⩽ i ⩽ 4, T_{1} − T_{2} = T_{R}, and T_{3} − T_{4} = T_{I}, so that T = T_{1} − T_{2} + i(T_{3} − T_{4}). Also, if T_{1}(f) = 0, then 0 > T_{R}(f) = −T_{2}(f), and therefore min(T_{1}, T_{2})(f) = T_{1}(f) = 0. If T_{1}(f) ≠ 0, then T_{1}(f) = max(T_{R}, 0)(f) = T_{R}(f) > 0, so that T_{2}(f) = 0 and min(T_{1}, T_{2})(f) = T_{2}(f) = 0. Thus min(T_{1}, T_{2}) = 0. Similarly min(T_{3}, T_{4}) = 0.

If T′_{i} satisfy the assumptions in the theorem above, then 0 = min(T′_{1}, T′_{2}) = T′_{1} − max(T′_{1} − T′_{2}, 0), and therefore T′_{1} = max(T′_{1} − T′_{2}, 0) = max(T_{R}, 0). It follows that T_{1} = T′_{1} and therefore T_{2} = T′_{2}. Similarly T_{3} = T′_{3} and T_{4} = T′_{4}. This proves our theorem.

At this point, we have set up all the machinery required to produce a measure from T ∈ C_{0}^{∗}. It is a standard fact in analysis^{3} that for any positive, linear functional I on C_{00}^{+}, there exists a measure ι defined on a σ-algebra of subsets of X containing the Borel σ-algebra such that for f ⩾ 0

∫_{X} fdι = \{ \begin{matrix}⟨f, I⟩, & f ∈ C_{00}^{+}; \\ sup ⟨g, I⟩ : g ∈ C_{00}^{+}, g ⩽ f & f ∈ 𝔐^{+}; \\ inf ⟨g, I⟩ : g ∈ 𝔐^{+}, f ⩽ g & otherwise. \end{matrix}  (11)

Since C_{00} is a dense linear subspace of C_{0}, the values of bounded linear functionals on C_{0} are completely determined by their values on C_{00}, so any bounded linear functional on C_{00} can be extended uniquely to a bounded linear functional on C_{0}. Thus in particular, the measure ι corresponding to I satisfies the equation

⟨f, I⟩ = ∫_{X} fdι

for all f ∈ C_{0}. We can then use the decomposition of a bounded linear functional into bounded positive linear functionals described above to construct a measure for a more general linear functional.

Theorem 21. Let T ∈ C_{0}^{∗}. Then there exists a measure μ ∈ M(X) such that T and μ satisfy Equation (7) with T_{μ} = T. In particular, Φ is surjective. Moreover, μ = μ_{1}−μ_{2}+i(μ_{3}−μ_{4}) with μ_{i} a finite, positive measure and Φ(μ_{i}) = T_{i} for 1 ⩽ i ⩽ 4, where the T_{i} are the uniquely defined bounded linear functionals from Theorem (20).

3This is a standard application of the Riesz Representation Theorem. See, for example, [3] pp.

212.

Proof. Let T_{i} with 1 ⩽ i ⩽ 4 be the uniquely defined positive, bounded linear functionals from Theorem (20). Then for each T_{i}, there exists a positive measure μ_{i} such that ⟨f, T_{i}⟩ = ∫_{X} fdμ_{i}. Since T_{i} is bounded, μ_{i} must be finite. Set μ = μ_{1} − μ_{2} + i(μ_{3} − μ_{4}). It follows that

∫_{X} fdμ = ∫_{X} fdμ_{1} − ∫_{X} fdμ_{2} + i (∫_{X} fdμ_{3} − ∫_{X} fdμ_{4})
= ⟨f, T_{1}⟩ − ⟨f, T_{2}⟩ + i (⟨f, T_{3}⟩ − ⟨f, T_{4}⟩) = ⟨f, T ⟩.

This proves our theorem.

1.4. Decomposition Relationships

This section is devoted to a couple of results which are interesting but inessential to the further development of the theory. The main reason we bring them up at this point is to provide a better idea what our decomposition above actually did, at least in terms of measures and the usual decompositions of measures. We will show in particular that if Φ : μ ↦ T_{μ}, then |μ| ↦ |T_{μ}| and we will relate the decomposition of T in Theorem (20) to the Jordan decomposition of a signed measure. The first result has already been obtained, but has yet to be formally mentioned.

Theorem 22. Let μ ∈ M(X) and T ∈ C_{0}^{∗}(X) with Φ : μ ↦ T. Then Φ : |μ| ↦ |T |.

Proof. This is simply a restatement of Equation (6).

The second result will require two results from measure theory.

Theorem 23 (Hahn Decomposition). Let μ be a signed measure on a σ-algebra Ω of subsets of X. A positive subset of X is a μ-measureable set E such that μ(F) ⩾ 0 for all F ⊂ E. A negative subset of X is a μ-measureable set E such that μ(F) ⩽ 0 for all F ⊂ E. Then there exists a positive set X_{+} and negative set X_{−} such that X_{+} ∪ X_{−} = X, X_{+} ∩ X_{−} = 0, μ(X_{+}) > 0, and μ(X_{−}) < 0. The choices of X_{−} and X_{+} are unique up to sets of measure 0.

Proof. See, for example, [3] pp. 86-87.

Theorem 24. (Jordan Decomposition) For a signed measure μ, there exist unique positive measures μ^{+} and μ^{−} such that μ = μ^{+} − μ^{−} and μ^{+} ⊥ μ^{−}.

Proof. Let X = X_{+} ∪X_{−} be a Hahn decomposition for μ. Define μ^{+}(E) = μ(E ∩X_{+}) and μ^{−}(E) = −μ(E ∩ X_{−}) for any μ-measureable set E. Clearly μ^{+} and μ^{−} are positive measures on the same measure space as μ and and satisfy μ = μ^{+} − μ^{−} and μ^{+} ⊥ μ^{−}. Let ν^{+} and ν^{−} be two positive measures on the measure space of μ such that μ = ν^{+} − ν^{−} and E and F are two μ-measureable sets satisfying E ∩ F = ∅, E ∪ F = X, and ν^{−}(E) = ν^{+}(F) = 0. Then E and F is another Hahn decomposition for μ and therefore ν^{+}(A) = ν^{+}(A ∩ E) = μ(A ∩ E) = μ(A ∩ X_{+}) = μ^{+}(A). Thus ν^{+} = μ^{+}. Similarly ν^{−} = μ^{−}.

For signed measures, the total variation is usually defined in terms of the Jordan decomposition. In particular, |μ| is defined to be μ^{+} + μ^{−}. This is necessary, since not all signed measures are finite measures and our definition of the total variation applies only to finite measures. In the case of finite signed measures, this definition turns out to be no different than ours, as the next theorem shows.

Theorem 25. Let μ be a finite signed measure. Then |μ| = μ^{+} + μ^{−}, where μ^{+} and μ^{−} are the components of the Jordan decomposition of μ.

Proof. Let X_{+} and X_{−} be the positive and negative sets, respectively, of the Hahn decomposition of X. Let U ∈ 𝔅(X). Then for any partition P of U,

∑_{V ∈P} |μ(V)| = ∑_{V ∈P} |μ(X_{+} ∩ V) + μ(X_{−} ∩ V)| ⩽ ∑_{V ∈P}(|μ(X_{+} ∩ V)| + |μ(X_{−} ∩ V)|)

= ∑_{V ∈P}(μ^{+}(V) + μ^{−}(V)) = μ^{+}(U) + μ^{−}(U).

Since the partition was arbitrary, we may conclude that |μ|(V) ⩽ μ^{+}(V) + μ^{−}(V). Conversely, {X_{+} ∩ U, X_{−} ∩ U} is a partition of U and therefore

μ^{+}(U) + μ^{−}(U) = |μ(X_{+} ∩ U)| + |μ(X_{−} ∩ U)| ⩽ |μ|(U).

Combined with the previous inequality, this proves μ^{+} + μ^{−} = |μ|.

Our next task will be to decompose a measure μ ∈ M(X) into real and imaginary components.

Theorem 26. Let μ ∈ M(X). For all Borel-measureable subsets U ∈ 𝔅(X), we define

μ_{R}(U) = Re(μ(U));  (12)
μ_{I}(U) = Im(μ(U)).  (13)

The real-valued set functions μ_{R} and μ_{I} are finite Borel measures on X (and therefore in M(X)).

Proof. We will prove the theorem for μ_{R}, since the proof for μ_{I} is similar. Let μ ∈ M(X). Since μ is a finite measure, it is clear that μ_{R}(U) ⩽ |μ(U)| < ∞ for all U ∈ 𝔅(X), and therefore μ_{R} is finite. Moreover, μ_{R}(∅) = Re(μ(∅)) = 0 and for any sequence of mutually disjoint sets {U_{i}}_{i=1}^{∞} ⊂ 𝔅(X), we have that

μ_{R} (⋃_{i=1}^{∞} U_{i}) = Re (μ (⋃_{i=1}^{∞} U_{i})) = Re (∑_{i=1}^{∞} μ(U_{i})) = ∑_{i=1}^{∞} Re(μ(U_{i})) = ∑_{i=1}^{∞} μ_{R}(U_{i}).

This proves that μ_{R} is a measure.

It follows from this theorem that any complex-valued measure μ may be expressed as a linear combination μ = μ_{R} + iμ_{I} of it’s real and imaginary components. Combining this with the Jordan decompositions of μ_{R} and μ_{I}, we may express μ as a sum of positive measures μ = μ_{R}^{+} − μ_{R}^{−} + i(μ_{I}^{+} − μ_{I}^{−}). Moreover, this decomposition is unique in the sense that the Jordan decompositions of μ_{R} and μ_{I} are unique. The next theorem shows that this decomposition is directly related to the decomposition of T ∈ C_{0}^{∗}(X) given by Theorem (20).

Theorem 27. Let T ∈ C_{0}^{∗}(X) and μ ∈ M(X) with Φ : μ ↦ T. Let T = T_{1} − T_{2} + i(T_{3} − T_{4}) be the unique decomposition of T given by Theorem (20). Then T_{1} = Φ(μ_{R}^{+}), T_{2} = Φ(μ_{R}^{−}), T_{3} = Φ(μ_{I}^{+}) and T_{4} = Φ(μ_{I}^{−}).

Proof. Define T_{R} and T_{I} by T_{R}(f) = Re(T (f)) and T_{I} = Im(T (f)) for all real, positive f ∈ C_{0}(X). Then, as before, T_{R} and T_{I} are real-valued, bounded linear functionals on C_{0}(X). Moreover, it is easy to see that T_{R} = μ_{R} and T_{I} = μ_{I}. Moreover, |μ_{R}| = μ_{R}^{+} + μ_{R}^{−}, so that |T_{R}| = Φ(μ_{R}^{+} + μ_{R}^{−}) and

T_{1} = max(T_{R}, 0) = \frac{1}{2}(T_{R} + |T_{R}|) = \frac{1}{2}(Φ(μ_{R}^{+} − μ_{R}^{−}) + Φ(μ_{R}^{+} + μ_{R}^{−})) = Φ(μ_{R}^{+}).

It follows that T_{2} = T_{1} − T_{R} = Φ(μ_{R}^{+}) − Φ(μ_{R}^{+} − μ_{R}^{−}) = Φ(μ_{R}^{−}). The proof that T_{3} = Φ(μ_{I}^{+}) and T_{4} = Φ(μ_{I}^{−}) is similar.

It is important to note that it is not true that |μ| = μ_{R}^{+} + μ_{R} + μ_{I}^{+} + μ_{I}^{−} unless μ_{R} ⊥ μ_{I}. This is simply a consequence of the fact that in general |T_{1}+T_{2}| ≠ |T_{1}|+|T_{2}| for T_{1}, T_{2} ∈ C_{0}^{∗}(X).

Before ending this section, we consider one last result which asserts that the isomorphism Φ is order-preserving. Theorem 28. Let μ_{i} ∈ M(X) be positive measures and let T_{i} ∈ C_{0}(X)^{∗} with μ_{i} = Φ(T_{i}) for i = 1, 2. Then μ_{1} ⩽ μ_{2} if and only if T_{1} ⩽ T_{2}. Proof. If μ_{1} ⩽ μ_{2}, then for any real-valued, positive function f ∈ C_{0}(X), we have that

T_{1}(f) = ∫_{X} fdμ_{1} ⩽ ∫_{X} fdμ_{2} = T_{2}(f),

and therefore T_{1} ⩽ T_{2}.

Conversely, assume that T_{1} ⩽ T_{2}, and let E ∈ 𝔅(X). Then there exists a monotone increasing sequence of real-valued, positive functions {f_{i}}_{i=1}^{∞} ⊂ C_{0}(X) such that f_{i} → 1_{E}. It follows from the monotone convergence theorem that

lim_{i→∞} T_{j}(f_{i}) = lim_{i→∞} ∫_{X} f_{i}dμ_{j} = ∫_{X} 1_{E}dμ_{j} = μ_{j}(E)

for j = 1, 2. Since T_{1} ⩽ T_{2}, T_{1}(f_{i}) ⩽ T_{2}(f_{i}) for all i. It follows that μ_{1}(E) ⩽ μ_{2}(E). This proves our theorem.

1.5. Product Measures and Fubini’s Theorem

Let X_{1} and X_{2} be locally compact Hausdorff spaces and ι_{1} and ι_{2} positive, σ-finite Borel measures on X_{1} and X_{2}, respectively. Then X_{1} × X_{2} is also a locally compact Hausdorff space under the product topology, and we define the product measure ι_{1}×ι_{2} in the usual way:

ι_{1} × ι_{2}(U) =  (14)

inf {∑_{i=1}^{n} μ(A_{i})μ(B_{i}) : A_{i} ∈ 𝔅(X_{1}), B_{i} ∈ 𝔅(X_{2}), U ⊂ ⋃_{i=1}^{n} A_{i} × B_{i} disjoint union } .

The measure ι_{1} × ι_{2} is a Borel measure on X_{1} × X_{2}, and ι_{1} × ι_{2}(A × B) = ι_{1}(A)ι_{2}(B) for any A ∈ 𝔅(X_{1}) and B ∈ 𝔅(X_{2}). Since ι_{1} and ι_{2} are σ-finite, so too is ι_{1} × ι_{2}, and it is the unique σ-finite measure satisfying ι_{1} × ι_{2}(A × B) = ι_{1}(A)ι_{2}(B). Moreover, ι_{1} × ι_{2}(X_{1} × X_{2}) = ι_{1}(X_{1})ι_{2}(X_{2}), so ι_{1} × ι_{2} is finite if and only if ι_{1} and ι_{2} are both finite. Before continuing, we recall an important result from measure theory. Theorem 29 (Fubini). Suppose that ι_{1} and ι_{2} are positive, σ-finite measures on X_{1} and X_{2}, respectively, and let f ∈ L^{1}(ι_{1} × ι_{2}). Define f(·, y) : X_{1} ↦ ℂ and f(x, ·) : X_{2} ↦ ℂ by f(·, y) : x ↦ f(x, y) and f(x, ·) : y ↦ f(x, y). Then f(·, y) ∈ L_{1}(ι_{1}) for ν-a.e. y and f(x, ·) ∈ L_{1}(ι_{2}) for μ-a.e. x. The functions y ↦ ∫ f(·, y)dμ and x ↦ ∫ f(x, ·)dν are defined ι_{2} and ι_{1}-a.e., respectively. Define F_{1}(x) to be ∫ f(x, y)dι_{2}(y) whenever the integral exists and 0 otherwise. Define F_{2}(y) to be ∫ f(x, y)dι_{1}(x) whenever the integral exists and 0 otherwise. Then F_{1} ∈ L_{1}(ι_{1}) and F_{2} ∈ L_{2}(ι_{2}) and

∫_{X×Y} fdι_{1} × ι_{2} = ∫_{X} F_{1}(x)dι_{1}(x) = ∫_{X_{2}} F_{2}(y)dι_{2}(y).

Proof. See, for example, [3] pp. 64-67.

Since a measure μ ∈ M(X) can be expressed as a linear combination of finite, positive measures, we will be able to define the product of complex measures. Definition 30. Let μ ∈ M(X_{1}) and ν ∈ M(X_{2}). Define μ_{1} = μ_{I}^{+}, μ_{2} = μ_{R}^{−}, μ_{3} = μ_{I}^{−}, and μ_{4} = μ_{R}^{+}. Also define ν_{1} = ν_{I}^{+}, ν_{2} = ν_{R}^{−}, ν_{3} = ν_{I}^{−}, and ν_{4} = ν_{R}^{+}. Then μ = ∑_{j=1}^{4}(i)^{j}μ_{j} and ν = ∑_{j=1}^{4}(i)^{j}ν_{j}. We define μ × ν to be the Borel measure on X_{1} × X_{2} given by

μ × ν = ∑_{j,k=1}^{4}(i)^{j+k}μ_{j} × ν_{k}.  (15)

Since μ × ν is a linear combination of finite measures, it too must be finite, so μ × ν ∈ M(X × Y). From the definition, for any A ∈ 𝔅(X_{1}) and B ∈ 𝔅(X_{2}) we have the identity

μ(A × B) = ∑_{j,k=1}^{4}(i)^{j+k}μ_{j} × ν_{k}(A × B) = ∑_{j,k=1}^{4}(i)^{j+k}μ_{j}(A)ν_{k}(B) = μ(A)ν(B).

The product of complex measures also preserves the total variation in a certain sense.

Theorem 31. Let μ ∈ M(X_{1}) and ν ∈ M(X_{2}). Then |μ × ν| = |μ| × |ν|.

Proof. Let A ∈ 𝔅(X_{1}) and B ∈ 𝔅(X_{2}), and consider 1_{A×B}(x, y) = 1_{A}(x)1_{B}(y). By Fubini’s theorem,

∫X_{1}×X_{2}1_{A×B}(x, y)\frac{dμ}{d|μ|}(x)\frac{dν}{d|ν|}(y)d|μ| × |ν|(x, y)

= ∫_{X_{1}} 1_{A}(x)\frac{dμ}{d|μ|}(x)d|μ|(x) ∫_{X_{2}} 1_{B}(y)\frac{dν}{d|ν|}(y)d|ν|(y)

= μ(A)ν(B) = ∫_{X_{1}×X_{2}} 1_{A×B}dμ × ν.

Define g(x, y) = \frac{dμ}{d|μ|}(x)\frac{dν}{d|ν|}(y). Then |g(x, y)| = 1 for |μ|-a.e. x and |ν|-a.e. y. It follows |g(x, y)| = 1 for |μ| × |ν|-a.e (x, y). Since the algebra of all sets of the form A × B with A ∈ 𝔅(X_{1}) and B ∈ 𝔅(X_{2}) generates 𝔅(X_{1} × X_{2}), we may conclude dμ × ν = gd|μ| × |ν|. It follows from Lemma (9) that d|μ × ν| = |g|d|μ| × |ν| = d|μ| × |ν|.

Using this result, we can prove a complex verision of Fubini’s theorem.

Theorem 32 (Fubini). Let μ ∈ M(X_{1}) and ν ∈ M(X_{2}), and f ∈ L_{1}(|μ × ν|). Define f(·, y) : X_{1} ↦ ℂ and f(x, ·) : X_{2} ↦ ℂ by f(·, y) : x ↦ f(x, y) and f(x, ·) : y ↦ f(x, y). Then f(·, y) ∈ L_{1}(|μ|) for ν-a.e. y and f(x, ·) ∈ L_{1}(|ν|) for μ-a.e. x. The functions y ↦ ∫ f(·, y)dμ and x ↦ ∫ f(x, ·)dν are defined ν and μ-a.e., respectively. Define F_{1}(x) to be ∫ f(x, y)dν(y) whenever the integral exists and 0 otherwise. Define F_{2}(y) to be ∫ f(x, y)dμ(x) whenever the integral exists and 0 otherwise. Then F_{1} ∈ L_{1}(|μ|) and F_{2} ∈ L_{2}(|ν|) and

∫_{X_{1}×X_{2}} fdμ × ν = ∫_{X} F_{1}(x)dμ(x) = ∫_{X_{2}} F_{2}(y)dν(y).

Proof. All the statements in the theorem are simply results of Fubini’s theorem for positive measures applied to |μ| and |ν|, except for the last one. From the proof of Theorem (31), we know that dμ × ν = gd|μ| × |ν| with g(x, y) = \frac{dμ}{d|μ|}(x)\frac{dν}{d|ν|}(y). Thus by Fubini’s theorem

∫_{X_{1}×X_{2}} fdμ × ν = ∫_{X_{1}×X_{2}} f(x, y)\frac{dμ}{d|μ|}(x)\frac{dν}{d|ν|}(y)d|μ| × |ν|(x, y)

= ∫_{X_{1}} ∫_{X_{2}} f(x, y)\frac{dμ}{d|μ|}(x)\frac{dν}{d|ν|}(y)d|μ|(x)d|ν|(y)

= ∫_{X_{1}} ∫_{X_{2}} f(x, y)dμ(x)dν(y).

Similarly, ∫_{X_{1}×X_{2}} fdμ × ν = ∫_{X_{1}} ∫_{X_{2}} f(x, y)dν(y)dμ(x).

1.6. The ∗-Algebra M(G)

Let G be a T_{0} topological group, and let F(G) be the collection of all complex-valued functions on G. Then G has continuous left and right translation operations, which map Borel sets to Borel sets. These operations give rise to a natural convolution operations on the linear space C_{0}^{∗}(G). Moreover, using inversion we may define a natural adjoint operation on C_{0}^{∗}(G). Thus we may consider C_{0}^{∗}(G), and by extension M(G), as a normed ∗-algebra. In this section, we develop the theory necessary to prove M(G) is a ∗-algebra. In the next section, we pick up the topic of the adjoint operation on M(G). We begin with some basic definitions.

Definition 33. An algebra A over a field k is a vector space over k with an additional binary operation of vector multiplication A × A → A : (x, y) ↦ x ∗ y, which satisfies the properties

(i) (x + y) ∗ z = x ∗ z + y ∗ z

(ii) z ∗ (x + y) = z ∗ x + z ∗ y

(iii) (αx) ∗ (βy) = αβ(x ∗ y)

for all x, y, z ∈ A and α, β ∈ k.

Unless otherwise stated, by an algebra we will always mean an algebra over the field of complex numbers ℂ.

Definition 34. Let A be an algebra. A surjective mapping x ↦ x^{∗} on A satisfying the properties

(i) (x + y)^{∗} = x^{∗} + y^{∗}

(ii) (αx)^{∗} = \overline{α}x^{∗}

(iii) (xy)^{∗} = y^{∗}x^{∗}

(iv) x^{∗∗} = x

for all x, y ∈ A and α ∈ ℂ is called an adjoint operation on A. An algebra with an adjoint operation is called a ∗-algebra. An element x of a ∗-algebra satisfying x^{∗} = x is called Hermitian. If A is a normed algebra with an adjoint operation and ∥x^{∗}∥ = ∥x∥ for all x ∈ A, then A is called a normed ∗-algebra. A Banach algebra that is a normed ∗-algebra is a Banach ∗-algebra.

To define the convolution operation, we first require some preliminary results.

Definition 35. For any function f ∈ F(G) and a ∈ G, define L(a)f to be the function defined by L(a)f(x) = f(a^{−1}x)^{4}. For any a ∈ G, the function L(a) is a linear operator on the complex vector space F(G). The function L : G → Hom(F(G), F(G)) is called the left regular representation of F(G).

Since L(ab)f(x) = f(b^{−1}a^{−1}x) = L(a)f(b^{−1}x) = L(a)L(b)f(x) for all a, b ∈ G, the left regular representation is a homomorphism of G into Hom(F(G), F(G)). For any a ∈ G the translation operation x ↦ a^{−1}x is continuous. Thus the restriction of L(a) to C(G), C_{0}(G), or C_{00}(G) is a linear operator on that linear subspace.

Lemma 36. Let T ∈ C_{0}^{∗}(G). Then for any f ∈ C_{0}(G), the function x ↦ ⟨L(x^{−1})f, T ⟩ is in C_{0}(G).

Proof. Let T ∈ C_{0}^{∗}(G) and f ∈ C_{0}. We must verify two things: the function defined above is continuous, the function can be made arbitrarily small outside a compact set. We begin by proving continuity. For all ϵ > 0, there exists an open neighborhood U of the identity e ∈ G such that for all x, y ∈ G, xy^{−1} ∈ U implies |f(x) − f(y)| < ϵ/∥T ∥. It follows that ∥L(x^{−1})f − L(y^{−1})f∥_{u} < ϵ/∥T ∥, and therefore |⟨L(x^{−1})f, T ⟩ − ⟨L(y^{−1})f, T ⟩| = |⟨L(x^{−1})f − L(y^{−1})f, T ⟩| < ϵ. This proves continuity.

Let ϵ > 0, and let μ ∈ M(G) be the measure satisfying Φ(μ) = T. To complete the proof, we must show that {x : |f(x)| ⩾ ϵ} is contained in a compact set. There exist compact subsets C_{1}, C_{2} ⊂ G such that |f(x)| < ϵ/2∥μ∥ for all x ∉ C_{1} and |μ|(C_{2}^{c}) < ϵ/2∥f∥_{u}. It follows that

|⟨L(a^{−1})f, T ⟩| = |∫_{G} f(ax)dμ(x)| ⩽ ∫_{G} |f(ax)|d|μ|(x) < ∫_{C_{2}} |f(ax)|d|μ|(x) + ϵ/2

If a ∉ C_{1}C_{2}^{−1} and x ∈ C_{2}, then ax ∉ C_{1} and therefore |f(ax)| < ϵ/2∥μ∥. Thus the above inequality tells us |⟨L(a^{−1})f, T ⟩| < ϵ for a ∉ C_{1}C_{2}^{−1}. Since the mapping x ↦ x^{−1} is continuous, C_{2}^{−1} is compact. The product of compact sets is compact, so C_{1}C_{2}^{−1} is compact. This proves that the function defined above is less than ϵ outside a compact set. Since ϵ > 0 was arbitrary, this proves our theorem.

4This operation may look a little strange at first: why not instead define L(a)f(x) = f(ax), as this would seem to be the simpler and more intuitive thing to do. The answer is that L would not be a homomorphism under this definition.

Using this lemma, we can now define the convolution of two linear functionals in C_{0}^{∗}(G).

Definition 37. Let T_{1}, T_{2} ∈ C_{0}^{∗}(G). The convolution of functionals T_{1} ∗ T_{2} is the linear functional on C_{0}(G) defined by

⟨f, T_{1} ∗ T_{2}⟩ = ⟨⟨L(·^{−1})f, T_{2}⟩, T_{1}⟩,

where ⟨L(·^{−1})f, T_{2}⟩ is the function in C_{0}(G) defined by x ↦ ⟨L(x^{−1})f, T_{2}⟩.

The next theorem shows that the convolution of functionals is a linear functional and that C_{0}^{∗}(G) is an algebra

Theorem 38. Let T_{1}, T_{2} ∈ C_{0}^{∗}(G). Then T_{1} ∗ T_{2} ∈ C_{0}^{∗}(G) and ∥T_{1} ∗ T_{2}∥ ⩽ ∥T_{1}∥∥T_{2}∥. The complex linear space C_{0}^{∗}(G) is an algebra with respect to this convolution operation.

Proof. For any f ∈ C_{0}(G) and T ∈ C_{0}^{∗}(G), let ⟨L(·^{−1})f, T ⟩ be the function in C_{0}(G) defined by x ↦ ⟨L(x^{−1})f, T_{2}⟩ and let L(·^{−1})f be the function defined by x ↦ L(x^{−1})f. Let f_{1}, f_{2}, f ∈ C_{0}(G) and α ∈ C. Then L(·^{−1})(αf) = αL(·^{−1})f and L(·^{−1})(f_{1} + f_{2}) = L(·^{−1})f_{1} + L(·^{−1})f_{2}. It follows that ⟨αL(·^{−1})f, T ⟩ = α⟨L(·^{−1})f, T ⟩, and ⟨L(·^{−1})(f_{1} + f_{2}), T ⟩ = ⟨L(·^{−1})f_{1}, T ⟩ + ⟨L(·^{−1})f_{2}, T ⟩. The linearity of T_{1} ∗ T_{2} follows immediately. Moreover

⟨⟨L(·^{−1})f, T_{2}⟩, T_{1}⟩ ⩽ ∥T_{1}∥∥⟨L(·^{−1})f, T_{2}⟩∥_{u} ⩽ ∥T_{1}∥∥T_{2}∥∥f∥_{u},

and therefore T_{1} ∗ T_{2} ∈ C_{0}^{∗}(G) with ∥T_{1} ∗ T_{2}∥ ⩽ ∥T_{1}∥∥T_{2}∥.

Additionally for any T ∈ C_{0}^{∗}(G) and any α ∈ ℂ, ⟨L(·^{−1})f, (T +T_{2})⟩ = ⟨L(·^{−1})f, T ⟩+ ⟨L(·^{−1})f, T_{2}⟩, and ⟨L(·^{−1})f, αT_{2}⟩ = α⟨L(·^{−1})f, T_{2}⟩. Thus

⟨f, T ∗ (T_{1} + T_{2})⟩ = ⟨⟨L(·^{−1})f, (T_{1} + T_{2})⟩, T ⟩ = ⟨⟨L(·^{−1})f, T_{1}⟩ + ⟨L(·^{−1})f, T_{2}⟩, T ⟩
= ⟨⟨L(·^{−1})f, T_{1}⟩, T ⟩ + ⟨⟨L(·^{−1})f, T_{2}⟩, T ⟩ = ⟨f, T ∗ T_{1}⟩ + ⟨f, T ∗ T_{2}⟩

and

T_{1} ∗ (αT_{2}) = ⟨⟨L(·^{−1})f, αT_{2}⟩, T_{1}⟩ = ⟨α⟨L(·^{−1})f, T_{2}⟩, T_{1}⟩ = α⟨⟨L(·^{−1})f, T_{2}⟩, T_{1}⟩

Similarly (T_{1} + T_{2}) ∗ T = T_{1} ∗ T + T_{2} ∗ T and (αT_{1}) ∗ T_{2} = αT_{1} ∗ T_{2}. It follows that C_{0}^{∗}(G) is an algebra. This proves our theorem.

The isometry Φ and the convolution on C_{0}^{∗}(G) gives us a readily available means of defining the convolution of measures.

Definition 39. Let μ, ν ∈ M(G), and let Φ(μ) = T_{μ} and Φ(ν) = T_{ν}. We define the convolution of measures μ∗ν to be the measure in M(G) such that Φ(μ∗ν) = T_{μ} ∗T_{ν}.

Theorem 40. Let μ_{1}, μ_{2} ∈ M(G). Then |μ_{1} ∗ μ_{2}| ⩽ |μ_{1}| ∗ |μ_{2}|.

Proof. Let T_{i} ∈ C_{0}^{∗}(G) be the linear functional corresponding to μ_{i} for i = 1, 2. Let f ∈ C_{0}(G) be a real-valued, positive function on G and let φ ∈ C_{0}(G) with |φ| ⩽ f. Then we have that

|⟨φ, T_{1} ∗ T_{2}⟩| = |∫_{G} ∫_{G} φ(xy)dμ_{2}(y)dμ_{1}(x)| ⩽ ∫_{G} |∫_{G} φ(xy)dν(y)| d|μ|(x)
⩽ ∫_{G} ∫_{G} |φ(xy)|d|ν|(y)d|μ|(x)
⩽ ∫_{G} ∫_{G} f(xy)d|ν|(y)d|μ|(x) = ⟨f, |T_{1}| ∗ |T_{2}|⟩.

Taking the supremum over all φ ∈ C_{0}(G) with |φ| ⩽ f, we find |T_{1} ∗ T_{2}| ⩽ |T_{1}| ∗ |T_{2}|. The isomorphism Φ is order-preserving, and therefore |μ_{1} ∗ μ_{2}| ⩽ |μ_{1}| ∗ |μ_{2}|.

The next theorem provides an alternative definition of the convolution of measures which turns out to be equivalent. Moreover, it describes for us a means of evaluating ∫_{G} fdμ_{1} ∗ μ_{2} in the case that f ∈ L_{1}(|μ_{1}| ∗ |μ_{2}|).

Theorem 41. Let μ_{1}, μ_{2} ∈ M(G) and let η be the Borel measure on G defined by η(E) = μ_{1} × μ_{2}(τ^{−1}(E)) for all E ∈ 𝔅(G). Then η = μ_{1} ∗ μ_{2} and for any Borelmeasureable function f with f ∘ τ ∈ L_{1}(|μ_{1}| × |μ_{2}|),

∫_{G} fdμ_{1} ∗ μ_{2} = ∫_{G} fdη = ∫_{G} f ∘ τdμ_{1} × μ_{2}  (16)
= ∫_{G} ∫_{G} f(xy)dμ_{2}(y)dμ_{1}(x) = ∫_{G} ∫_{G} f(xy)dμ_{1}(x)dμ_{2}(y).

Moreover, f ∈ L_{1}(|μ_{1}| ∗ |μ_{2}|) implies f ∘ τ ∈ L_{1}(|μ_{1}| × |μ_{2}|).

Proof. Let T_{i} ∈ C_{0}^{∗}(G) be the linear functional corresponding to μ_{i} for i = 1, 2. If f ∈ C_{0}(G), then f ∘τ ∈ C_{0}(G), and therefore f ∘τ ∈ L_{1}(|μ_{1}|×|μ_{2}|). Thus by Fubini’s theorem,

T_{1} ∗ T_{2}(f) = ∫_{G} ∫_{G} f(xy)dμ_{2}(y)dμ_{1}(x) = ∫_{G×G} f ∘ τdμ_{1} × μ_{2}.

The mapping f ↦ ∫_{G×G} f ∘ τdμ_{1} × μ_{2} is a linear functional on C_{0}(G). Moreover, by a similar argument, Fubini’s theorem tells us

|∫_{G×G} f ∘ τdμ_{1} × μ_{2}| ⩽ ∫_{G×G} |f ∘ τ|d|μ_{1} × μ_{2}|
= ∫_{G×G} |f| ∘ τd|μ_{1}| × |μ_{2}| = ∫_{G×G} |f| ∘ τd|μ_{1}| × |μ_{2}|
= ∫_{G} |f|d|μ_{1}| ∗ |μ_{2}| = |T_{1}| ∗ |T_{2}|(|f|) ⩽ ∥T_{1}∥∥T_{2}∥∥f∥_{u}.

It follows that f ↦ ∫_{G×G} f ∘ τdμ_{1} × μ_{2} is a bounded linear functional on C_{0}(G), and therefore is the same as T_{1} ∗ T_{2}. Now given any E ∈ 𝔅(G), τ^{−1}(E) ∈ 𝔅(G × G), since τ is continuous. Also, τ(x, y) ∈ E if and only if (x, y) ∈ τ^{−1}(E). It follows that 1_{E} ∘ τ = 1_{τ^{−1}(E)} and therefore η(E) = ∫_{G} 1_{E} ∘ τdμ_{1} × μ_{2} = ∫_{G} 1_{τ^{−1}(E)}dμ_{1} × μ_{2} = μ_{1} × μ_{2}(τ^{−1}(E)). From this, we may conclude that the equality

∫_{G} f ∘ τdμ_{1} × μ_{2} = ∫_{G} fdη

holds for all simple Borel measureable functions f, and therefore for all Borel measureable functions f with f ∘ τ ∈ L_{1}(|μ_{1}| × |μ_{2}|). Since Φ is an isomorphism, we may conclude η = μ_{1} ∗ μ_{2}. This combined with Fubini’s theorem proves Equation (16).

Next, consider a compact subset C ⊂ G, and let ϵ > 0. Then there exists an open subset U ⊂ G such that C ⊂ U and |μ|∗|ν|(U)−|μ|∗|ν|(C) < ϵ. Let f ∈ C_{0}(G) be a function with support in U, with f(x) = 1 for all x ∈ C, and with 0 ⩽ f ⩽ 1. Then by Fubini’s theorem

∫_{G×G} 1_{C} ∘ τd|μ_{1}| × |μ_{2}| = ∫_{G} ∫_{G} 1_{C}(xy)d|μ_{2}|(y)d|μ_{1}|(x)
⩽ ∫_{G} ∫_{G} f(xy)d|μ_{2}|(y)d|μ_{1}|(x) = |T_{1}| ∗ |T_{2}|(f)
= ∫_{G} fd|μ_{1}| ∗ |μ_{2}| ⩽ ∫_{G} 1_{U}d|μ_{1}| ∗ |μ_{2}|
= |μ_{1}| ∗ |μ_{2}|(U) < |μ_{1}| ∗ |μ_{2}|(C) + ϵ.

Since ϵ > 0 was arbitrary, it follows that the inequality

∫_{G×G} 1_{C}fd|μ_{1}| × |μ_{2}| ⩽ |μ_{1}| ∗ |μ_{2}|(C),

for all compact sets, and therefore for all σ-compact Borel sets. Moreover, if O ∈ 𝔅(G) is a set of |μ_{1}| ∗ |μ_{2}|-measure zero, then for any compact subset C ⊂ τ^{−1}(O), τ(C) is compact and 1_{C} ⩽ 1_{τ(C)} ∘ τ. It follows that

|μ_{1}| × |μ_{2}|(C) ⩽ ∫_{G} 1_{τ(C)} ∘ τd|μ_{1}| × |μ_{2}| ⩽ |μ_{1}| ∗ μ_{2}|(C) ⩽ |μ_{1}| ∗ |μ_{2}|(O) = 0.

Since this is true of any compact subset of τ^{−1}(O), it follows that

∫_{G} 1_{O} ∘ τd|μ_{1}| × |μ_{2}| = |μ_{1}| × |μ_{2}|(τ^{−1}(O)) = 0.

Now given any E ∈ 𝔅(G), since |μ| ∗ |ν| is σ-finite, E may be written as the union of disjoint sets E = C ∪ O, where C is σ-compact and O is a set of |μ| ∗ |ν|-measure 0. It follows that

∫_{G} 1_{E} ∘ τd|μ_{1}| × |μ_{2}| = ∫_{G} 1_{C} ∘ τd|μ_{1}| × |μ_{2}| + ∫_{G} 1_{O} ∘ τd|μ_{1}| × |μ_{2}|
⩽ |μ| ∗ |ν|(C) + |μ| ∗ |ν|(O) = |μ| ∗ |ν|(E).

The inequality

∫_{G} f ∘ τd|μ_{1}| × |μ_{2}| ⩽ ∫_{G} fd|μ| ∗ |ν|(E)

then follows immediately for simple functions and subsequently for positive Borelmeasureable functions by the monotone convergence theorem. This in turn guarantees it for all f ∈ L_{1}(|μ| ∗ |ν|), proving our theorem.

The above theorem provides us with the most direct means of actually evaluating the convolution of two measures, as the next examples demonstrate.

Example 42. Let G = R and let a > 0. Let μ be the measure defined by dμ = \frac{1}{2a}1_{[−a,a]}dx, where dx represents the usual Lebesgue measure on R. Then by Theorem (41) below and Fubini’s theorem,

μ ∗ μ((−∞, b]) = ∫_{R×R} 1_{(−∞,b]}(x + y)dμ(x) × dμ(y)

= \frac{1}{4a^{2}} ∫_{−a}^{a} ∫_{−a}^{a} 1_{(−∞,b]}(x + y)dydx

= \frac{1}{4a^{2}} ∫_{−a}^{a} ∫_{x−a}^{x+a} 1_{(−∞,b]}(y)dydx

= \{ \begin{matrix}0, & b ⩽ −2a \\ \frac{1}{8a^{2}} (2a + b)^{2}, & −2a ⩽ b ⩽ 0; \\ 1 − \frac{1}{8a^{2}} (2a − b)^{2}, & 0 ⩽ b ⩽ 2a; \\ 1, & b ⩾ 2a. \end{matrix}

It follows that dμ ∗ μ = (1 − |x|/(2a))1_{[−2x,2x]}(x)dx. Example 43. Let G be the group of all 2 × 2 matrices of the form

(x, y) := ( \begin{matrix}x & y \\ 0 & x^{−1} \end{matrix} ) ,

with x ≠ 0. Then (a, b)(x, y) = (ax, ay + bx^{−1}) and G is a topological group under the metric defined by ρ((x, y), (x′, y′)) = \sqrt{(x − x′)^{2} + (y − y′)^{2}} and has a left Haar measure λ defined by dλ(x, y) = \frac{1}{x^{2}} dm(x, y), where m is the usual Lebesgue measure on (ℝ\{0}) × ℝ. If f ∈ L_{1}(λ), then the measure μ defined by dμ = fdλ is a complex measure absolutely continuous with respect to λ. Let δ_{(a,b)} be the dirac measure at (a, b). Then dδ_{(a,b)} ∗ μ = L((a, b))fdλ = f((a^{−1}, −b)(x, y))dλ(x, y).

1.7. Decomposition of M(G) and Adjoint Operators

In this section, we decompose the space M(G) into a direct sum of linear subspaces and in particular prove that the collection M_{a}(G) of measures absolutely continuous with respect to a left Haar measure λ on G is an ideal in M(G). Thus, given any μ ∈ M(G) and ν ∈ M_{a}(G), the measure μ ∗ ν ∈ M_{a}(G), and therefore there exists a function g such that dμ ∗ ν = gdλ. In particular, since M_{a}(G) is isometric to L_{1}(λ), the mapping f ↦ μ ∗ f, where μ ∗ f ∈ L_{1}(G) is the function defined by dμ ∗ ν = μ ∗ fdλ with dν = fdλ, is a bounded linear operator on L_{1}(G). Moreover, we show that M(G) ∗ L_{p}(λ) ⊂ L_{p}(λ) for all 1 ⩽ p ⩽ ∞, and therefore the mapping f ↦ μ ∗ f is a bounded linear operator on the Hilbert space L_{2}(λ). In particular then, this demands the existence of an adjoint, which we shall define.

As a vector space, the algebra M(G) can be decomposed into a direct sum of linear subspaces. To begin, we have the following definitions.

Definition 44. A measure μ ∈ M(G) is said to be continuous if μ({x}) = 0 for all x ∈ G, and the collection of all such measures is denoted by M_{c}(G). It is said to be discrete if the support of the measure is contained in a countable set, and the collection of all discrete measures is denoted M_{d}(G). It is absolutely continuous if it is continuous with respect to a left Haar measure on G, and the collection of all such measures is denoted M_{a}(G). Lastly, a measure is singular if it is continuous and singular with respect to a left Haar measure on G, and the collection of all such measures is denoted M_{s}(G).

It follows immediately from the definitions that M_{a}(G), M_{d}(G), and M_{s}(G) are all linear subspaces of M(G) which intersect only at {0}. Moreover, we have the following theorem.

Theorem 45. If G is not discrete, then M(G) can be decomposed into the linear sum

M(G) = M_{d}(G) ⊕ M_{s}(G) ⊕ M_{a}(G).

Thus, each μ ∈ M(G) may be decomposed into a direct sum μ = μ_{d} + μ_{s} + μ_{a} of discrete, singular and absolutely continuous measures and

|μ| = |μ_{d}| + |μ_{s}| + |μ_{a}|

and

∥μ∥ = ∥μ_{d}∥ + ∥μ_{s}∥ + ∥μ_{a}∥.

Proof. Given μ ∈ M(G), the collection A_{μ} of all x ∈ G such that μ({x}) ≠ 0 is countable^{5}, since μ is finite. Define μ_{d} ∈ M(G) by μ_{d}(E) = μ(E ∩ A_{μ}) for all E ∈ 𝔅(G). Then μ_{d} ∈ M_{d}(G) and μ_{c} = μ−μ_{d} is continuous. Moreover the Lebesgue-Radon-Nikodym Theorem tells us that μ_{c} = μ_{s} + μ_{a}, where μ_{s} is singular and μ_{a} is absolutely continuous. It follows that μ = μ_{d} + μ_{s} + μ_{a} ∈ M_{d}(G) + M_{s}(G) + M_{a}(G). Since M_{a}(G), M_{d}(G), and M_{s}(G) pairwise intersect to {0}, this proves M(G) = M_{d}(G) ⊕ M_{s}(G) ⊕ M_{a}(G).. The measures μ_{d}, μ_{a}, and μ_{s} are all mutually singular by definition, and therefore |μ| = |μ_{d}| + |μ_{s}| + |μ_{a}|. The rest of the theorem follows immediately.

In order to prove that the convolution of a measure in μ ∈ M(G) with a measure ν of the form dν = fdλ with f ∈ L_{p}(G) is again a measure of the form dμ ∗ ν = gdλ for some g ∈ L_{p}(G), we require a technical lemma.

Lemma 46. Let f be a λ-measureable function on G. Then (x, y) ↦ f(x^{−1}y) and (x, y) ↦ f(x) are λ × |μ|-measureable for all μ ∈ M(G)

Proof. See [2] pp. 287-88.

Using this lemma, we have the following theorem.

5By a countable set, we mean a set which is either finite or countably infinite.

Theorem 47. Suppose that 1 ⩽ p, q < ∞ with 1/p + 1/q = 1 and that f ∈ L_{p}(λ). Let κ(x, y) = x^{−1}y and μ ∈ M(G). Then if p < ∞, the function h defined by

μ ∗ f(x) = ∫_{G} f(y^{−1}x)dμ(y)  (17)

exists and is finite for λ-a.e. x ∈ G. Moreover μ ∗ f ∈ L_{p}(λ), and ∥h∥_{p} ⩽ ∥f∥_{p}∥μ∥. If p = 1 and ν ∈ M(G) is the measure defined by dν = fdλ, then dμ ∗ ν = μ ∗ fdλ. Proof. By our previous lemma, we know that f ∘ κ is λ × |μ|-measureable. Hölder’s inequality tells us that for all h ∈ L_{q}(λ),

∫_{G} |f(x^{−1}y)h(y)|dy ⩽ ∥L(x^{−1})f∥_{p}∥h∥_{q} = ∥f∥_{p}∥h∥_{q},

from which it follows that

∫_{G} ∫_{G} |f(x^{−1}y)h(y)|dyd|μ|(x) ⩽ ∫_{G} ∥f∥_{p}∥h∥_{q}d|μ|(x) = ∥f∥_{p}∥h∥_{q}∥μ∥.

Thus

∫_{G} ∫_{G} f(x^{−1}y)h(y)\frac{dμ}{d|μ|}(x)d|μ|(x)dy = ∫_{G} h(y) (∫_{G} f(x^{−1}y)\frac{dμ}{d|μ|}(x)d|μ|(x)) dy

is a well-defined function in L_{1}(λ) and it follows that

h(y) ∫_{G} f(x^{−1}y)\frac{dμ}{d|μ|}(x)d|μ|(x)

exists and is finite for λ-a.e. y ∈ G. We define

μ ∗ f(y) = ∫_{G} f(x^{−1}y)\frac{dμ}{d|μ|}(x)d|μ|(x) = ∫_{G} f(x^{−1}y)dμ(x)

whenever the integral exists and 0 otherwise. Since the value of h was arbitrary, necessarily μ ∗ f is λ-measureable.

Since f ∈ L_{p}(λ), the support of f is contained in a σ-compact subset A ⊂ G. Moreover, since μ ∈ M(G), there exists a σ-compact subset B ⊂ G such that μ(B^{c}) = 0. Let C = BA. Then since the product of compact sets is compact, C is σ-compact. Thus there exists a sequence of functions {ψ_{i}}_{i=1}^{∞} ⊂ C_{00}(G) with 0 ⩽ ψ_{i} ⩽ 1 for all i and ψ_{i}(x) → 1_{C}(x) for λ-a.e. x ∈ G.

μ ∗ f(y) = lim_{n→∞} ψ_{i}(y) ∫_{G} f(x^{−1}y)\frac{∂μ}{∂|μ|}(x)d|μ|(x)

exists and is finite and equal to μ ∗ f(y) for λ-a.e. y ∈ C. Moreover, from the definition of μ ∗ f, if y ∉ C then x^{−1}y ∉ A for all x ∈ B and therefore μ ∗ f(y) = 0. Thus the support of μ ∗ f is contained in a compact set and

∥μ ∗ f∥_{p} = sup {|∫_{G} μ ∗ f(y)h(y)dy| : h ∈ L_{q}(G), ∥h∥_{q} ⩽ 1}
< ∥f∥_{p}∥h∥_{q}∥μ∥ ⩽ ∥f∥_{p}∥μ∥.

Thus μ ∗ f ∈ L_{q}(λ) and ∥μ ∗ f∥_{p} ⩽ ∥f∥_{p}∥μ∥.

Lastly, let ν ∈ M(G) be the measure defined by dν = fdλ. Then μ ∗ f ∈ L_{1}(λ). Moreover, for any φ ∈ C_{00}(G) Fubini’s theorem tells us that

∫_{G} φμ ∗ fdλ = ∫_{G} ∫_{G} φ(y)f(x^{−1}y)dμ(x)dy = ∫_{G} ∫_{G} φ(y)f(x^{−1}y)dydμ(x)

= ∫_{G} ∫_{G} φ(xy)f(y)dydμ(x) = ∫_{G} ∫_{G} φ(xy)dν(y)dμ(x) = ∫_{G} φdμ ∗ ν..

Since this is true for every φ ∈ C_{00}(G), the corresponding measures correspond to the same linear functional on C_{0}(G). Thus we have that μ ∗ fdλ = dμ ∗ ν. This proves our theorem.

The previous theorem tells us immediately that M_{a}(G) is a left ideal of M(G) and allows us to accept the following definition.

Definition 48. Let μ ∈ M(G) and f_{1}, f_{2} ∈ L_{p}(λ) for some 1 ⩽ p < ∞. Moreover, let ν_{i} ∈ M(G) be the measure defined by dν_{i} = f_{i}dλ for i = 1, 2. If p = 1, we define μ ∗ f_{1}, f_{1} ∗ f_{2} ∈ L_{1}(λ) to be the functions satisfying μ ∗ ν_{1} = μ ∗ f_{1}dλ and ν_{1} ∗ ν_{2} = f_{1} ∗ f_{2}dλ. If p ≠ 1, then we define μ ∗ f as in Equation (17) of Theorem (47).

Let μ ∈ M(G). The above theorem shows us that the function T : f ↦ μ ∗ f is a linear operator on L_{2}(G). Since this operator acts on a Hilbert space, we expect there to be an adjoint operator T^{∗} such that ⟨T f, h⟩ = ⟨f, T^{∗}h⟩ for all f, h ∈ L_{2}(λ). In particular, this means that

∫_{G} f(x)\overline{(T^{∗}g)(x)}dx = ∫_{G} μ ∗ f(x)\overline{g(x)}dx = ∫_{G} ∫_{G} f(y^{−1}x)dμ(y)\overline{g(x)}dx
= ∫_{G} ∫_{G} f(y^{−1}x)\overline{g(x)}dxdμ(y) = ∫_{G} ∫_{G} f(x)\overline{g(yx)}dxdμ(y)
= ∫_{G} f(x) ∫_{G} \overline{g(yx)}dμ(y)dx.

Since the value of f ∈ L_{2}(λ) was arbitrary, we may conclude that

\overline{T^{∗}g(x)} = ∫_{G} \overline{g(yx)}dμ(y).

Recall that θ : G → G is the map defined by θ(x) = x^{−1}. Suppose that there is a measure μ^{∗} ∈ M(G) such that

∫_{G} fdμ^{∗} = \overline{∫_{G} \overline{f ∘ θ}dμ}.

for all f ∘ θ ∈ L_{1}(|μ|). Then

μ^{∗} ∗ f(x) = ∫_{G} f(y^{−1}x)dμ^{∗}(y) = \overline{∫_{G} \overline{f(θ(y^{−1})x)}dμ^{∗}(y)} = T^{∗}f,

and we are able to successfully express the adjoint operator T^{∗} as a convolution with a measure. As it turns out, for all μ ∈ M(G) there is a measure μ^{∗} ∈ M(G), satisfying this property, and this defines an adjoint on M(G) (and simultaneously on C_{0}^{∗}(G)).

Definition 49. Let μ ∈ M(G) and let T ∈ C_{0}^{∗}(G) with Φ(μ) = T. We define the adjoint of T to be the linear functional T^{∗} ∈ C_{0}^{∗}(G) given by T^{∗}(φ) = \overline{T (\overline{φ ∘ θ})} for all φ ∈ C_{0}(G). We define the adjoint of μ to be the measure μ^{∗} satisfying Φ(μ^{∗}) = T^{∗}.

Since the composition φ∘θ of a function φ ∈ C_{0}(G) with the continuous function θ is in C_{0}(G), it is obvious from the definition that T^{∗} is a bounded linear functional on C_{0}(G). Moreover, the adjoint operation T ↦ T^{∗} establishes C_{0}^{∗}(G) as a ∗-algebra. In particular, we have the following theorem.

Theorem 50. Under the operation adjoint operation, C_{0}^{∗}(G) is a normed ∗-algebra. Furthermore, |T |^{∗} = |T^{∗}| for all T ∈ C_{0}^{∗}(G).

Proof. Let T, T_{1}, T_{2} ∈ ℂ_{0}^{∗}(G). It is clear from the definition that (T_{1} +T_{2})^{∗} = T_{1}^{∗} +T_{2}^{∗}, (αT)^{∗} = \overline{α}T^{∗}, and T^{∗∗} = T. Given any φ ∈ C_{0}(G), φ∘θ ∈ C_{0}(G) with ∥φ∘θ∥_{u} = ∥φ∥_{u}. Moreover, φ ∘ θ ∘ θ = φ and therefore {φ : φ ∈ C_{0}(G), |φ| ⩽ 1} = {φ ∘ θ : φ ∈ C_{0}(G) : |φ| ⩽ 1\}. Moreover \{φ : φ ∈ C_{0}(G), |φ| ⩽ 1\} = \{\overline{φ} : φ ∈ C_{0}(G), |φ| ⩽ 1\}, and therefore \{φ : φ ∈ C_{0}(G), |φ| ⩽ 1\} = \{\overline{φ ∘ θ} : φ ∈ C_{0}(G) : |φ| ⩽ 1\}. It follows that

∥T^{∗}∥ = sup{∥T^{∗}(φ)∥_{u} : φ ∈ C_{0}(G), |φ| ⩽ 1}
= sup\{∥T (\overline{φ ∘ θ})∥_{u} : φ ∈ C_{0}(G), |φ| ⩽ 1\}
= sup{∥T (φ)∥_{u} : φ ∈ C_{0}(G), |φ| ⩽ 1} = ∥T ∥.

Therefore the adjoint operation perserves the norm on C_{0}^{∗}(G).

Let μ_{i} ∈ M(G) be the measures satisfying Φ(μ_{i}) = T_{i} for i = 1, 2. Given any φ ∈ C_{0}(G), Fubini’s theorem tells us that

(T_{1} ∗ T_{2})^{∗}(φ) = \overline{(T_{1} ∗ T_{2})(\overline{φ ∘ θ})} = \overline{∫_{G} ∫_{G} \overline{φ ∘ θ(xy)}dμ_{2}(y)dμ_{1}(x)}

= \overline{∫_{G} ∫_{G} \overline{φ(y^{−1}x^{−1})}dμ_{2}(y)dμ_{1}(x)} = \overline{∫_{G} ∫_{G} \overline{φ(y^{−1}x^{−1})}dμ_{1}(x)dμ_{2}(y)}

= \overline{∫_{G} ∫_{G} \overline{L(y^{−1})(φ ∘ θ)(x)}dμ_{1}(x)dμ_{2}(y)} = \overline{∫_{G} \overline{T_{1}^{∗}(L(y^{−1})φ)}dμ_{2}(y)}
= \overline{∫_{G} \overline{T_{1}^{∗}(L(θ(y))φ)}dμ_{2}(y)} = T_{2}^{∗} ∗ T_{1}^{∗}(φ).

Since ϕ was arbitrary, this shows us that (T_{1} ∗T_{2})^{∗} = T_{2}^{∗} ∗T_{1}^{∗}. This proves that ℂ_{0}^{∗}(G) is a normed ∗-algebra.

The last thing that we wish to show is |T |^{∗} = |T^{∗}|. For any real, positive function φ ∈ C_{0}(G) we have that |T |^{∗}(φ) = |T |(φ ∘ θ). Moreover if ψ ∈ C_{0}(G) with |ψ| ⩽ φ, then |\overline{ψ ∘ θ}| ⩽ |ψ ∘ θ| ⩽ φ ∘ θ. Thus

|T^{∗}|(φ) = sup{|T^{∗}(ψ)| : ψ ∈ C_{0}(G), |ψ| ⩽ φ}
= sup\{|T (\overline{ψ ∘ θ})| : ψ ∈ C_{0}(G), |ψ| ⩽ φ\}
⩽ sup{|T (ψ)| : ψ ∈ C_{0}(G), |ψ| ⩽ φ ∘ θ} = |T |(φ ∘ θ) = |T |^{∗}(φ).

It follows that |T^{∗}| ⩽ |T |^{∗}. It is also clear from the definition that for real, positive linear functionals T_{1}, T_{2}, T_{1} ⩽ T_{2} implies T_{1}^{∗} < T_{2}^{∗} and vise-versa. Consequently,

|T |^{∗} = |(T^{∗})^{∗}|^{∗} ⩽ |T^{∗}|^{∗∗} ⩽ |T^{∗}|.

Combining these inequalities, we conclude |T |^{∗} = |T^{∗}|.

As with all the other operations on C_{0}^{∗}(G), there is a strictly measure theoretic interpretation. This is discussed in the next theorem. In particular, the next theorem shows that μ^{∗} is the same measure we found during our search for the adjoint the linear transformation f ↦ μ ∗ f on L_{2}(λ).

Theorem 51. Let T ∈ C_{0}^{∗}(G) and let μ be the measure on M(G) with Φ(μ) = T. Then for every E ∈ 𝔅(G), we have μ^{∗}(E) = \overline{μ(θ(E))} and |μ^{∗}|(B) = |μ|(θ(B)). Moreover f ∈ L_{1}(|μ^{∗}|) if and only if f ∘ θ ∈ L_{1}(|μ|). In this case,

∫_{G} fdμ^{∗} = \overline{∫_{G} \overline{f ∘ θ}dμ}.

If f ∈ L_{p}(G) with 1 ⩽ p < ∞, then

μ^{∗} ∗ f(x) = \overline{∫_{G} \overline{f(yx)}dμ(y)}

for λ-a.e. x ∈ G.

Proof. Since μ^{∗} ∈ M(G), is finite, for any open subset U ⊂ G, there exists an increasing sequence of functions {ψ_{i}}_{i=1}^{∞} ⊂ C_{00}(G) with 0 ⩽ ψ_{i} ⩽ 1 and ψ_{i} → 1_{U} pointwise |μ^{∗}|-a.e.. Similarly, there exists an increasing sequence of functions {φ_{i}}_{i=1}^{∞} ⊂ C_{00}(G) with 0 ⩽ φ_{i} ⩽ 1 and ψ_{i} → 1_{θ(U)} pointwise |μ|-a.e.. Define ω_{i}(x) = max{ψ_{i}(x), φ_{i}(x^{−1})} for all x ∈ G. Then ω_{i} → 1_{U} pointwise |μ^{∗}|-a.e. and ω_{i}^{∗} → 1_{θ(U)} pointwise |μ|-a.e.. Thus by the dominated convergence theorem,

μ^{∗}(U) = lim_{i→∞} ∫_{G} ω_{i}dμ^{∗} = lim_{i→∞} \overline{∫_{G} \overline{ω_{i} ∘ θ}dμ} = lim_{i→∞} \overline{∫_{G} ω_{i}dμ} = \overline{μ(U^{−1})}.

Since this is true of all open sets, the identity μ^{∗}(E) = \overline{μ(θ(E))} follows immediately for all Borel sets E.

The identity ∫_{G} φdμ^{∗} = \overline{∫_{G} \overline{f ∘ θ}dμ} then follows immediately if φ is a simple function. If f ∈ L_{1}(|μ^{∗}|), then there exists a sequence of simple functions {φ_{i}}_{i=1}^{∞} such that φ_{i} → f pointwise |μ^{∗}|-a.e. and |φ_{i}| ⩽ |φ_{i+1}| ⩽ |f| for all i ⩾ 1. Thus the dominated convergence theorem tells us that

∫_{G} fdμ^{∗} = lim_{i→∞} φ_{i}dμ^{∗} = lim_{i→∞} \overline{∫_{G} \overline{φ_{i} ∘ θ}dμ}.

Moreover, since |T^{∗}| = |T |^{∗} for all T ∈ C_{0}^{∗}(G), we have that |μ^{∗}| = |μ|^{∗} and the monotone convergence theorem tells us that

∫_{G} |f ∘ θ|d|μ| = lim_{i→∞} ∫_{G} |φ_{i} ∘ θ|d|μ| = lim_{i→∞} ∫_{G} |φ_{i}|d|μ|^{∗}

= lim_{i→∞} ∫_{G} |φ_{i}|d|μ^{∗}| ⩽ ∫_{G} |f|d|μ^{∗}| < ∞.

Thus f ∘ θ ∈ L_{1}(|μ|) and the dominated convergence theorem tells us that

lim_{i→∞} \overline{∫_{G} \overline{φ_{i} ∘ θ}dμ} = \overline{∫_{G} \overline{f ∘ θ}dμ}.

Thus we have shown that if f ∈ L_{1}(|μ^{∗}|) then f ∘ θ ∈ L_{1}(|μ|) and

∫_{G} fdμ^{∗} = \overline{∫_{G} \overline{f ∘ θ}dμ}.

The converse follows from the fact that T^{∗∗} = T^{∗} for all T ∈ C_{0}^{∗}(G) and therefore μ^{∗∗} = μ, so that in particular if f∘θ ∈ L_{1}(|μ|) = L_{1}(|μ^{∗}∗|), then f = f∘θ∘θ ∈ L_{1}(|μ^{∗}|).

The remainder of the theorem then follows immediately from Theorem (50).

CHAPTER 2. REPRESENTATIONS OF LOCALLY COMPACT GROUPS

2.1. Introduction

The purpose of this chapter is to lay down the necessary theory for representations of a locally compact topological group G. In particular, we are interested in defining representations of groups, the continuity and measureability of representations, and establishing an important result attributed to Gel’fand and Raĭkov that shows that the collection of irreducible unitary representations of a locally compact topological group separates points in G. Throughout this chapter, G will denote a locally compact topological group, X will denote a Banach space, and by H we will mean a Hilbert space^{6}. Additionally, by B(X, X) and U(H) we will mean the collection of all bounded operators on X and all unitary operators on H, respectively. We also use X^{∗} to denote the dual space of X. As in the previous chapter, λ will denote a left Haar measure on G and we will use dx and dλ(x) interchangeably^{7}. In addition, we will use δ_{a} to denote the dirac measure at a ∈ G and e to denote the identity of G.

2.2. Basic Definitions and Facts

We begin with a rather large collection of definitions that will setup the framework for the results in this chapter.

Definition 52. We define a representation of a semigroup S as homomorphism π : S → Hom_{k}(V, V) of S into the semigroup of all linear operators on V, where V is a vector space over a field k. We define a representation of an algebra A in the same way, only now π is required to be an algebra homomorphism of A in to the algebra Hom_{k}(V, V) of linear operators on V. The vector space V is called the representation space of the representation π. A subspace U ⊂ V is invariant under π if it is invariant under all the linear operators in the image of π: π(a)(U) ⊂ U for all a ∈ A (or S).

6This notation is different from the previous chapter, where X was a locally compact Hausdorff space and G was used in place of X to represent a locally compact T_{0} topological group.

7If G is compact, λ will always be the unique Haar measure such that λ(G) = 1.

Unless otherwise mentioned, by algebra we will always mean an algebra over ℂ. Moreover, for k = ℂ we will denote Hom_{k}(V, V) as Hom(V, V). Moreover G will always refer to a locally compact group. Definition 53. Let V be a topological vector space over a field k, and let T ∈ Hom(V, V). A subspace U ⊂ V is invariant under T if T (U) ⊂ U. An artibrary collection {T_{α}}_{α∈Λ} ⊂ V is called reducible if there is a nontrivial closed proper subspace invariant under T_{α} for every α ∈ Λ and irreducible otherwise. A representation is called irreducible if {π(x)}_{x∈G} is irreducible.

In our exposition of unitary operators on compact groups, the algebra M(G) and the concept of the adjoint will play a role. The ∗-algebras that will be of interest to us are those which are subalgebras of M(G) closed under the adjoint operation μ ↦ μ^{∗}. Corresponding to ∗-algebras are ∗-representations. Definition 54. A ∗-representation of a ∗-algebra A is a representation π of A by bounded operators on a Hilbert space H satisfying π(a^{∗}) = π(a)^{∗} for all a ∈ A. A ∗-representation π is faithful if π(a) ≠ 0 for a ≠ 0. Definition 55. A representation π of an algebra A or a semigroup S over a Hilbert space H is called unitary if π(x) is unitary for all x ∈ G. If π_{1} and π_{2} are representations by bounded operators on Hilbert spaces H_{1} and H_{2} respectively, then π_{1} and π_{2} are equivalent if there exists a linear isometry T : H_{1} → H_{2} such that the diagram

\xymatrix{H_{1} \ar[r]^{π_{1}(a)} \ar[d]_{T} & H_{1} \ar[d]^{T} \\ H_{2} \ar[r]^{π_{2}(a)} & H_{2},}

commutes for all a ∈ A (or S). That is, T π_{1}(a) = π_{2}(a)T for all a ∈ A (or S). This defines an equivalence relation on G.

Definition 56. Let π be a representation of an algebra A (or semigroup S) by operators on a vector space V. Then π defines a (left) action of A (or S) on V. For any v ∈ V, we define the orbit v with respect to π as Orb(v, π) = {π(a)v : a ∈ A} (or {π(a)v : a ∈ S}). If V is a topological vector space, the representation π is said to be cyclic if there exists v ∈ V such that the subspace of V generated by Orb(v, π) is dense in V. An element v satisfying this property is said to be a cyclic vector of π.

Given a representation π of a group G (or algebra A) by operators on a Banach space X there will in general be several induced maps that will have properties necessary in considering questions of continuity and measureability. As a notational convenience, if F is a function whose domain is a subset of B(X, X), the collection of bounded operators on X we will denote the induced mapping with domain G by F (π(·)). To give a better idea of what we mean, we list a few examples that will recur frequently:

π(·) : G → B(X, X)  where x ↦ (π(·))(x) = π(x);  (18)
π(·)ψ : G → X  where x ↦ (π(·)ψ)(x) = π(x)ψ  (19)
⟨π(·)ψ, ϕ⟩ : G → ℂ  where x ↦ (⟨π(·)ψ, ϕ⟩)(x) = ⟨π(x)ψ, ϕ⟩,  (20)

where ψ ∈ X and ϕ ∈ X^{∗}. Using this notation, we establish our next collection of definitions.

Definition 57. Let X be a Banach space and π be a representation of G by bounded operators on X. The representation π is weakly Borel measureable if the function ⟨π(·)ψ, ϕ⟩ is weakly Borel measureable for all ψ ∈ X and ϕ ∈ X^{∗}. The representation is weakly continuous if the function ⟨π(·)ψ, ϕ⟩ is continuous for all ψ ∈ X and ϕ ∈ X^{∗}.

If the mapping π(·)ψ is continuous for all ψ ∈ X, the representation is called strongly continuous. If sup_{x∈G} ∥π(x)∥ = c < ∞, then π is called totally bounded.

The motivation for the definition of a totally bounded representation stems from the fact that such representations map G into totally bounded subsets of B(X, X).

2.3. Extensions of Representations to M(G)

Take a representation π of G over a reflexive Banach space X by members of B(X, X)^{8}. Then π immediately defines a representation of the subalgebra of M_{d}(G) in the sense that π maps the atom δ_{x} to π(x). In fact, if π has nice enough properties, we can extend π to a corresponding representation π_{A} of M(G) such that the diagram

\xymatrix{G \ar[r]^{π} \ar[d]_{g↦δ_{g}} & B(X, X) \\ M(G) \ar[ur]^{π_{A}}}

commutes. The properties required for π to be extended are stated in the next theorem.

Theorem 58. Let A be a subalgebra of M(G). Let π be a representation of G by bounded operators on a reflexive Banach space X such that

(a) π is weakly |μ|-measureable, and weakly |μ| ∗ |ν|-measureable for all μ, ν ∈ A;

(b) π is totally bounded and sup_{x∈G} ∥π(x)∥ = c.

Then for every μ ∈ A, there exists a unique operator π_{A}(μ) such that

⟨π_{A}(μ)f, h⟩ = ∫_{G}⟨π(x)f, h⟩dμ(x)  (21)

8A reflexive Banach space is a Banach space X where the dual space of the dual space X^{∗∗} is isometrically isomorphic to X. In this case, we use the convention X = X^{∗∗} and will let f ∈ X denote both itself and it’s counterpart in X^{∗∗} under the isometry.

for all f ∈ B and h ∈ B^{∗}. Moreover the mapping μ ↦ π_{A}(μ) is a representation of A by bounded operators on B such that ∥π_{A}(μ)∥ ⩽ c∥μ∥ for all μ ∈ A.

Proof. Let μ ∈ A. For any ψ ∈ X and ϕ ∈ X^{∗}, since ⟨π(x)ψ, ϕ⟩ is |μ|-measureable and |⟨π(x)ψ, ϕ⟩| ⩽ ∥π(x)ψ∥∥ϕ∥ ⩽ c∥ψ∥∥ϕ∥, we know that ⟨π(x)ψ, ϕ⟩ ∈ L_{1}(μ). Moreover for fixed ψ ∈ X and x ∈ G, the mapping

h ↦ ∫_{G}⟨π(x)αψ, ϕ⟩dμ(x)

is obviously conjugate-linear and

|∫_{G}⟨π(x)αψ, ϕ⟩dμ(x)| ⩽ ∫_{G} |⟨π(x)αψ, ϕ⟩|d|μ|(x) ⩽ c∥ψ∥∥ϕ∥|μ|(G) < ∞

so that it is necessarily bounded. Every bounded conjugate-linear functional on the reflexive Banach space X^{∗} is of the form ϕ ↦ ⟨ψ̃, h⟩ for some ψ̃ ∈ X^{∗∗} = X. Identifying ψ̃^{∗∗} with π_{A}(μ)ψ in the bounded conjugate-linear functional above defines a mapping π_{A}(μ) on X satisfying Equation (21). Let ψ_{1}, ψ_{2} ∈ X. For all ϕ ∈ X^{∗}, we have that

⟨π_{A}(μ)(ψ_{1} + ψ_{2}), h⟩ = ∫_{G}⟨π(x)(ψ_{1} + ψ_{2}), ϕ⟩dμ(x)
= ∫_{G}⟨π(x)ψ_{1}, ϕ⟩dμ(x) + ∫_{G}⟨π(x)ψ_{2}, ϕ⟩dμ(x)
= ⟨π_{A}(μ)ψ_{1}, ϕ⟩ + ⟨π_{A}(μ)ψ_{1}, ϕ⟩

which implies π_{A}(μ)(ψ_{1} +ψ_{2}) = π_{A}(μ)ψ_{1} +π_{A}(μ)ψ_{2}. Similarly π_{A}(μ)(αψ) = απ_{A}(μ)ψ for all ψ ∈ X and α ∈ ℂ, so that π_{A}(μ) is a linear transformation on X. The inequality |⟨π(μ)ψ, ϕ⟩| ⩽ c∥ψ∥∥ϕ∥|μ|(G) tells us

∥π_{A}(μ)∥ = sup_{0≠ψ∈B} \frac{∥π_{A}(μ)ψ∥}{∥ψ∥} = sup_{0≠ϕ∈X^{∗}} \frac{|⟨π_{A}(μ)ψ, ϕ⟩|}{∥ψ∥∥ϕ∥} ⩽ c|μ|(G) = c∥μ∥.

At this point, we have shows that μ ↦ π_{A}(μ) is a function from A into B(X, X). The linearity relations π_{A}(μ + ν) = π_{A}(μ) + π_{A}(ν) and π_{A}(αμ) = απ_{A}(μ) for all μ, ν ∈ A and α ∈ ℂ are immediate consequences of the definition of of π_{A}. Thus we need only verify the relation π_{A}(μ ∗ ν) = π_{A}(μ)π_{A}(ν) for all μ, ν ∈ A. For any ψ ∈ X and ϕ ∈ X^{∗}, we have that

⟨π_{A}(μ ∗ ν)ψ, ϕ⟩ = ∫_{G}⟨π(x)ψ, ϕ⟩d(μ ∗ ν)(x) = ∫_{G} ∫_{G}⟨π(xy)ψ, h⟩dν(y)dμ(x)
= ∫_{G} ∫_{G}⟨π(x)π(y)ψ, ϕ⟩dν(y)dμ(x)

= ∫_{G} ∫_{G}⟨π(y)ψ, (π(x))^{∗}ϕ⟩dν(y)dμ(x) = ∫_{G}⟨π_{A}(ν)ψ, (π(x))^{∗}ϕ⟩dμ(x)

= ∫_{G}⟨π(x)π_{A}(ν)ψ, ϕ⟩dμ(x) = ⟨π_{A}(μ)π_{A}(ν)ψ, ϕ⟩.

This proves our theorem.

In the above, we considered a subalgebra A of M(G). Clearly if A contains M_{d}(G), then π(δ_{g}) = π(g). To establish the requirements for the representation to be a ∗-representation, we first consider the “adjoint algebra” A^{∗} = {μ^{∗} : μ ∈ A}. For any subalgebra A of M(G), A^{∗} is also a subalgebra of A, and a representation π extendable to a representation of A is also extendable to a representation of A^{∗}. Moreover, if A is closed under adjoints, B is a Hilbert space, and the representation π is unitary, then the extension π_{A} is a ∗-representation of M(G). This is made clear by the following theorem. Theorem 59. Let A be a subalgebra of M(G) and π a representation of A by bounded operators on a reflexive Banach space X satisfying the assumptions of Theorem (58).

Then for every μ^{∗} ∈ A^{∗}, there exists a unique operator π̃_{A}(μ) such that

⟨π̃_{A}(μ^{∗})ϕ, ψ⟩ = ∫_{G}⟨(π(x^{−1}))^{∗}ϕ, ψ⟩dμ^{∗}(x) = ⟨ϕ, π_{A}(μ)ψ⟩  (22)

for all ϕ ∈ X^{∗} and ψ ∈ X^{∗∗} = X. For every μ ∈ A, we have π̃_{A}(μ^{∗}) = (π_{A}(μ))^{∗}. If X is a Hilbert space and we identify X = X^{∗} = H, and π is unitary, then π_{A}(μ) = π̃_{A}(μ), so that π is a ∗-representation of the ∗-algebra A.

Proof. Define a function π̃ mapping G to operators on X^{∗} via π̃ : x ↦ (π(x^{−1}))^{∗}. Since π is a representation of G by bounded operators on X, we see that

(π((xy)^{−1}))^{∗} = (π(y^{−1})π(x^{−1}))^{∗} = (π(x^{−1}))^{∗}(π(y^{−1}))^{∗},

and therefore π̃ is a representation of G by bounded operators on X^{∗}. Since

⟨π̃(x)ϕ, ψ⟩ = \overline{⟨ψ, (π(x^{−1}))^{∗}ϕ⟩} = \overline{⟨(π(x^{−1}))ψ, ϕ⟩},

property (a) of Theorem (58) holds for π̃. Furthermore, π̃(x^{−1}) = (π(x))^{∗} and ∥(π(x))∥ = ∥(π(x))^{∗}∥, and therefore property (b) of Theorem (58) holds for the representation π^{∗}. The previous theorem then grants us the existence of a representation π̃_{A} satisfying the left-hand side equality of Equation (22). The other equality in Equation (22) follows from a standard manipulation:

⟨π̃_{A}(μ^{∗})ϕ, ψ⟩ = ∫_{G}⟨(π(x^{−1}))^{∗}ϕ, ψ⟩dμ^{∗}(x) = ∫_{G}⟨ϕ, π(x^{−1})ψ⟩dμ^{∗}(x)
= ∫_{G} \overline{⟨π(x^{−1})ψ, ϕ⟩}dμ^{∗}(x) = \overline{∫_{G}⟨π(x)ψ, ϕ⟩dμ(x)} = ⟨ϕ, π_{A}(μ)ψ⟩.

In particular, this implies that (π_{A}(μ))^{∗} = π̃_{A}(μ^{∗}). The last part of the theorem is a consequence of the fact that for unitary operators π(x^{−1}) = (π(x))^{∗}, so that π̃(x^{−1}) = π(x) and therefore (π_{A}(μ))^{∗} = π̃_{A}(μ^{∗}) = π_{A}(μ).

An important question to try to answer at this point concerns the invariance of sets under the extension we have described above. In particular, we are interested in finding a general set of conditions under which a π invariant subspace of a Hilbert space H is μ_{A} invariant and visa-versa. As it turns out, if the representation π is unitary, the extension π_{A} defined by Theorem (58) respects invariance in the sense that subspaces of the corresponding Hilbert space invariant under π are also invariant under π_{A}. We have the converse when π is weakly continuous and for every open set U of G, A contains a nonzero positive measure supported on U. This is essentially the content of the next theorem.

Theorem 60. Let A be a ∗-subalgebra of M(G) and π a representation of G by unitary operators on a Hilbert space H satisfying (a) of Theorem (58) (since π is unitary, it automatically satisfies (b)). Then every closed subspace of H invariant under π is invariant under π_{A}. If π is weakly continuous as a representation of G and for every nonvoid open subset U of G there exists a nonnegative real measure μ ∈ A such that μ(U) = 1 and μ(U^{c}) = 0, then every closed subspace of H invariant under π_{A} is invariant under π.

Proof. Let V be a closed subspace of H invariant with respect to π and P be the projection of H onto V. Let T be any operator on H. Given any f ∈ H, we may write f as the linear combination ψ = ψ_{1} + ψ_{2} where ψ_{1} = P ψ and ψ_{2} = ψ − ψ_{1}. Clearly P ψ_{1} = ψ_{1} and P ψ_{2} = 0. It follows that T P ψ = T ψ_{1}, so that P T ψ = T P ψ if and only if P T ψ = T ψ_{1}. Thus V is invariant with respect to an operator T on H if and only if P commutes with T. It follows that P π(x) = π(x)P for all x ∈ G. Recall that P^{∗} = P. Then for all μ ∈ A and ψ, ϕ ∈ H, we have that

⟨π_{A}(μ)P ψ, ϕ⟩ = ∫_{G}⟨π(x)P ψ, ϕ⟩dμ(x) = ∫_{G}⟨P π(x)ψ, ϕ⟩dμ(x)
= ∫_{G}⟨π(x)ψ, P ϕ⟩dμ(x) = ⟨π_{A}(μ)ψ, P ϕ⟩ = ⟨P π_{A}(μ)ψ, ϕ⟩.

Therefore π_{A}(μ)P = P π_{A}(μ) for all μ ∈ A. This proves that V is invariant with respect to π_{A}.

To prove the second part of the theorem, suppose that V is a closed subspace of H invariant with respect to π_{A}, and let P be the projection of H onto V. Fix ψ, ϕ ∈ H. From a previous calculation, we know that

⟨P π_{A}(μ)ψ, ϕ⟩ = ∫_{G}⟨P π(x)ψ, ϕ⟩dμ(x).

Since π_{A}P = P π_{A}, we have that

0 = ⟨P π_{A}(μ)ψ, ϕ⟩ − ⟨π_{A}P ψ, ϕ⟩ = ∫_{G}⟨(P π(x) − π(x)P)f, h⟩dμ(x)

for all μ ∈ A. Since π is weakly continuous, the function ⟨(P π(x) − π(x)P)ψ, ϕ⟩ is a continuous bounded function of x. Thus if there exists an a ∈ G with ⟨(P π(a) − π(a)P)ψ, ϕ⟩ ≠ 0, then there must be a nonempty open subset U of G such that the real or imaginary portion of ⟨(P π(a)−π(a)P)ψ, ϕ⟩ does not change sign on U. Taking μ to be the positive measure on U with μ(U) = 1 and μ(U^{c}) = 0, we find that the integral

∫_{G}⟨(P π(x) − π(x)P)ψ, ϕ⟩dμ(x) = ∫_{U}⟨(P π(x) − π(x)P)ψ, ϕ⟩dμ(x) ≠ 0,

which is a contradiction. We conclude that ⟨(P π(x) − π(x)P)f, h⟩ = 0 for all x ∈ G.

Since ψ, ϕ ∈ H was arbitrary, this proves that π(x) and P commute. Thus V is invariant with respect to π.

2.4. Continuity of Representations

In this section, we discuss the continuity of representations. Specifically, we obtain the result that absolutely bounded weakly continuous representations are strongly continuous. These representations turn out to have the utility we need, and we simply call them “continuous representations”. Theorem 61. Let π be a weakly λ-measureable representation of G by bounded operators on a reflexive Banach space X that is absolutely bounded, and suppose the function ⟨π(·)f, h⟩ is continuous at e for all ψ ∈ X and ϕ ∈ X^{∗}. Then the function π(·)ψ is a left uniformly continuous mapping of G into X with respect to the norm topology on X. In particular, every weakly continuous totally bounded representation of G is strongly continuous. Proof. Since π is λ-measureable, π property (a) of Theorem (58) for A = M_{a}(G). Extend π to a representation π_{A} of M_{a}(G) as in Equation (21) of Theorem (58). We first show that each element ψ ∈ X is contained in the closure of its orbit with respect to π_{A}. This will be done by contradiction. Thus assume that ψ ∈ X such that ψ ∉ \overline{Orb(ψ, π_{A})}. Then by the Hahn-Banach Theorem (see [6]), there exists a linear functional ϕ ∈ X^{∗} with ⟨ψ, ϕ⟩ = 1 and ⟨Orb(ψ, π_{A}), ϕ⟩ = {0}. Fix a positive constant 0 < c < 1. Since ⟨π(e)ψ, ϕ⟩ = 1 and ⟨(π(·)ψ, ϕ⟩ is continuous at e, there exists a neighborhood U of the identity e ∈ G with λ(U) > 0 such that Re⟨π(·)ψ, ϕ⟩ > c on U. If we let μ ∈ M_{a}(G) be the measure such that dμ = 1_{U}dλ, then

Re⟨π(μ)ψ, ϕ⟩ = ∫_{G} Re⟨π(x)ψ, ϕ⟩1_{U}dx = ∫_{U} Re⟨π(x)ψ, ϕ⟩dx > cλ(U) > 0.

This is a contradiction. We conclude that ψ ∈ \overline{Orb(ψ, π_{A})} for all ψ ∈ X.

Define b = sup_{x∈G} ∥π(x)∥. Consider any a ∈ G, ψ ∈ X, ϕ ∈ X^{∗}, and μ ∈ M_{a}(G). A quick calculation shows

⟨π(δ_{a} ∗ μ)ψ, ϕ⟩ = ∫_{G}⟨π(x)ψ, ϕ⟩d(δ_{a} ∗ μ)(x) = ∫_{G} ∫_{G}⟨π(xy)ψ, ϕ⟩δ_{a}(x)dμ(y)
= ∫_{G}⟨π(ay)ψ, ϕ⟩dμ(y) = ∫_{G}⟨π(a)π(y)ψ, ϕ⟩dμ(y)

= ∫_{G}⟨π(y)ψ, π(a)^{∗}ϕ⟩dμ(y) = ⟨π_{A}(μ)ψ, π(a)^{∗}ϕ⟩ = ⟨π(a)π_{A}(μ)ψ, ϕ⟩.

In particular,

|⟨π(a)π_{A}(μ)ψ − π_{A}(μ)ψ, ψ⟩| ⩽ |⟨(π_{A}(δ_{a} ∗ μ) − π_{A}(μ))ψ, ϕ⟩| ⩽ b∥ψ∥∥ϕ∥δ_{a} ∗ μ − μ∥.

Since μ ≪ λ, d(δ_{a} ∗μ) = L(a)\frac{dμ}{dλ}dλ. Since x ↦ L(x)\frac{dμ}{dλ} is a right uniformly continuous function from G into L_{1}(G), for all ϵ > 0, there exists a neighborhood U_{μ} of e in G such that

∥δ_{a} ∗ μ − μ∥ = |δ_{a} ∗ μ − μ|(G) = ∫_{G} |(L(a)\frac{dμ}{dλ}) (x) − \frac{dμ}{dλ}(x)| dx

= ∥L(a)\frac{dμ}{dλ} − \frac{dμ}{dλ}∥_{1} < \frac{ϵ}{2∥ψ∥b^{2}}

for all a ∈ U_{μ}. Therefore

∥π(a)π_{A}(μ)ψ − π_{A}(μ)ψ∥ ⩽ \frac{ϵ}{2b}

for all a ∈ U_{μ}. Since ψ is in the closure of the Orb(ψ, π_{A}), we may choose μ ∈ M_{a}(G) such that

(1 + b)∥ψ − π_{A}(μ)ψ∥ < \frac{ϵ}{2b}.

Then we have that for all x ∈ U_{μ}

∥π(x)ψ − ψ∥ ⩽ ∥π(x)ψ − π(x)π_{A}(μ)ψ∥ + ∥π(x)π_{A}(μ)ψ − π_{A}(μ)ψ∥ + ∥π_{A}(μ)ψ − ψ∥

⩽ b∥ψ − π_{A}(μ)ψ∥ + \frac{ϵ}{2b} + ∥π_{A}(μ)ψ − ψ∥ < \frac{ϵ}{2b} + \frac{ϵ}{2b} = \frac{ϵ}{b}.

Thus, if x, y ∈ G with y^{−1}x ∈ U_{μ}, we find

∥π(x)ψ − π(y)ψ∥ = ∥π(y)(π(y^{−1}x)ψ − ψ)∥ ⩽ ∥π(y)∥∥π(y^{−1}x)ψ − ψ∥ < ϵ.

This proves our theorem.

In light of this theorem, we make the following definition.

Definition 62. Let π be a representation of G by operators on a reflexive Banach space X under which the mapping π(·)ψ is a left uniformly continuous mapping of G onto X with respect to the topology defined by the norm of X. Then π is called a continuous representation of G. In particular, weakly λ-measureable, weakly continuous or strongly continuous, and absolutely bounded representations are continuous representations of (G).

2.5. Restriction of Representations of M(G)

Given a representation π of G, we have found some general conditions under which we may extend to a representation π_{A} of a ∗-subalgebra A of M(G). The use of “extend” entails that we are considering G as a subset of A, via the correspondence g ↦ δ_{g}. To be more accurate, π_{A} is an extension of a representation of the ∗-subalgebra of A generated by the atoms δ_{g}. There is an obvious problem with this: the ∗-algebra A may or may not contain the point masses. We have an alternative when A is closed under the operation of convolution with point masses and the representation π_{A} satisfies certain nice properties. This is the topic of the next theorem.

Theorem 63. Let A be a Banach ∗-subalgebra of M(G) such that δ_{g} ∗ μ ∈ A for all g ∈ G and μ ∈ A, and let π_{A} be a ∗-representation of A by bounded operators on a Hilbert space H such that for all ψ ∈ H there exists μ ∈ M(G) such that π_{A}(μ)ψ ≠ 0. Then there exists a representation π of G by unitary operators on H.

Proof. We can decompose the Hilbert space H in to a direct sum H = N ⊕⨁_{γ∈Γ} H_{γ} of π_{A}-invariant subspaces of H, where π_{A}(μ)(N) = {0} for all μ ∈ Γ and the restriction of the representation π_{A} to H_{γ} is cyclic. By assumption N = {0}. Let P_{γ} be the projection operator of H onto H_{γ}. Suppose that for each γ we have a unitary representation π_{γ} of G by unitary operators on H_{γ} satisfying the relation

⟨π_{A}(μ)P_{γ}ψ, P_{γ}ϕ⟩ = ∫_{G}⟨π_{γ}(x)P_{γ}ψ, P_{γ}ϕ⟩dμ(x)

for all ψ, ϕ ∈ H. For any a ∈ G, define π(a) to be the operator on H defined by

π(a)ψ = ⨁_{γ∈Γ} π_{γ}(a)P_{γ}.

This is a direct sum of mutually orthogonal unitary operators, and therefore defines a unitary operator on H. Moreover this orthogonality gives us the identity

π(xy) = ⨁_{γ∈Γ} π_{γ}(xy)P_{γ} = ⨁_{γ∈Γ} π_{γ}(x)π_{γ}(y)P_{γ} = ⨁_{γ,γ′∈Γ} π_{γ}(x)P_{γ}π_{γ′}(y)P_{γ} = π(x)π(y),

and therefore π is a unitary representation of G. From its definition, π satisfies Equation (21).

Thus we need only prove our theorem for the special case that π_{A} is a cyclic representation. With this in mind, we assume that π_{A} is a cyclic ∗-representation of A satisfying the properties of the theorem. Then there exists a vector φ ∈ H such that Orb(φ, π_{A}) is dense in H. Using the fact that δ_{x}^{∗} = δ_{x^{−1}} and π_{A} is a ∗-algebra representation, we have the relation [π_{A}(δ_{x} ∗ μ)]^{∗} = π_{A}(μ^{∗} ∗ δ_{x}^{∗}) = π_{A}(μ^{∗} ∗ δ_{x^{−1}}) for all μ ∈ A. Thus

∥π_{A}(δ_{x} ∗ μ)φ∥^{2} = ⟨π_{A}(δ_{x} ∗ μ)φ, π_{A}(δ_{x} ∗ μ)φ⟩ = ⟨f, [π_{A}(δ_{x} ∗ μ)]^{∗}π_{A}(δ_{x} ∗ μ)φ⟩
= ⟨φ, π_{A}(μ^{∗} ∗ δ_{x^{−1}})π_{A}(δ_{x} ∗ μ)φ⟩ = ⟨φ, π_{A}(μ^{∗} ∗ δ_{x^{−1}} ∗ δ_{x} ∗ μ)φ⟩
= ⟨φ, π_{A}(μ^{∗} ∗ μ)φ⟩ = ⟨φ, [π_{A}(μ)]^{∗}π_{A}(μ)φ⟩ = ⟨π_{A}(μ)φ, π_{A}(μ)φ⟩
= ∥π_{A}(μ)φ∥^{2}

for all μ ∈ A and x ∈ G. For every x ∈ G, define an operator π′(x) on Orb(φ, π_{A}) by π′(x)(π_{A}(μ)φ) = π_{A}(δ_{x} ∗ μ). If μ, ν ∈ A satisfy π_{A}(μ)φ = π_{A}(ν)φ, then the above calculation shows that

∥π′(x)(π_{A}(μ)φ) − π′(x)(π_{A}(ν)φ)∥ = ∥π_{A}(δ_{x} ∗ μ)φ − π_{A}(δ_{x} ∗ ν)φ∥ =
∥π_{A}(δ_{x} ∗ (μ − ν))φ∥ = ∥π_{A}(μ − ν)φ∥ = ∥π_{A}(μ)φ − π_{A}(ν)φ∥ = 0.

Thus the mapping π′(x) is well-defined. Moreover, for any μ, ν ∈ A and α ∈ ℂ, we have that

π′(x)(π_{A}(μ)φ + π_{A}(ν)φ) = π′(x)(π_{A}(μ + ν)φ) = π_{A}(δ_{x} ∗ (μ + ν))φ

= π_{A}(δ_{x} ∗ μ + δ_{x} ∗ ν)φ = π_{A}(δ_{x} ∗ μ)φ + π_{A}(δ_{x} ∗ ν)φ = π′(x)(π_{A}(μ)φ) + π′(π_{A}(ν)φ)

and similarly π′(x)(π_{A}(αμ)φ) = απ′(x)(π_{A}(μ)φ). Moreover, we have already shown that π′(x) is norm-preserving. We therefore conclude that π′(x) is a linear isometry on Orb(φ, π_{A}) for all x ∈ G. Since Orb(φ, π_{A}) is dense in H, this representation π′(x) can be extended to a unique linear isometry (and therefore unitary operator) π(x) on H. Since δ_{xy} = δ_{x} ∗ δ_{y}, it is evident that π′(xy) = π′(x)π′(y), from which it immediately follows that π(xy) = π(x)π(y). Hence π is a representation of G by unitary operators on H. This proves our theorem.

The generality of the conditions under which we can restrict a representation π_{A} of A to a representation π of G is good news. However, without knowing more about the properties of the representation π_{A} it would be difficult to discuss the properties of π. In particular, we will not know in general if π is continuous, absolutely bounded, or even measureable. In the next theorem, we consider some conditions under which a representation of a ∗-subalgebra A of M(G) can be pulled back to a representation of G with certain nice properties.

Theorem 64. Let A be a Banach ∗-subalgebra of M(G) and π_{A} a representation of A by bounded operators on a Hilbert space H satisfying the assumptions of Theorem (63). Additionally suppose every bounded linear functional on A has the form μ ↦ ∫_{G} l(x)dμ(x) for some bounded function l on G such that for every σ-compact subset S ⊂ G, 1_{S}l is Borel measureable. Then there is a representation π of G by unitary operators on H that is weakly μ and μ ∗ ν-measureable for all μ, ν ∈ A satisfying Equation (21).

Proof. We have already shown the existence of a unitary representation π in Theorem (63), so we need only consider the measureability condition and also show that this representation satisfies Equation (21). Using the notation of the previous theorem, to show that π is weakly |μ|-measureable, note that ⟨π(g)f, h⟩ = ⨁_{γ∈Γ}⟨π_{γ}(g)P_{γ}f, h⟩. Thus if each of the π_{γ} is weakly |μ|-measureable, ⟨π(g)f, h⟩ is a countable sum of |μ|- measureable functions and therefore measureable. It follows that π is |μ|-measureable. Thus we need only consider the special case when H is cyclic, with cyclic vector φ. Let h ∈ H. Any ∗-representation π_{A} is bounded (see [2], pp. 320), so the mapping μ ↦ ⟨ρ(μ)φ, h⟩ is a bounded linear functional on A for every h ∈ H. Thus by assumption, there exists a function l such that

⟨ρ(μ)φ, ϕ⟩ = ∫_{G} l(x)dμ(x)

for all μ ∈ A. Moreover

⟨π(g)π_{A}(μ)φ, h⟩ = ⟨π_{A}(δ_{g} ∗ μ)φ, h⟩ = ∫_{G} l(x)d(δ_{g} ∗ μ)(x)
= ∫_{G} ∫_{G} l(xy)dδ_{g}(x)dμ(y) = ∫_{G} l(gy)dμ(y).

Let μ and ν be any measures on M(G). Since μ and ν are finite, we may choose σ-compact subsets S_{μ}, S_{ν} ⊂ G such that |μ| is supported on S_{μ} and |ν| is supported on S_{ν}. It follows that |μ| × |ν|((S_{μ} × S_{ν})^{c}) = 0. Let τ : G × G → G be the binary operation defined by τ(x, y) = xy. Since S_{μ} × S_{ν} ⊂ τ^{−1}(S_{μ}S_{ν}), the identity l(xy) = (h ∘ τ)(x, y) = (l ∘ τ)(x, y)1_{S_{μ}S_{ν}}(x, y) = l(xy)1_{S_{μ}S_{ν}} holds |μ| × |ν|-a.e.. Since l1_{S_{μ}S_{ν}} is Borel measureable by assumption, we have that l ∘ τ ∈ L_{1}(|μ| × |ν|) and therefore the function

x ↦ ∫_{G} l(xy)dν(y) = ⟨π(x)π_{A}(ν)φ, ϕ⟩

is |μ|-measureable for all μ ∈ M(G). Since Orb(φ, π_{A}) is dense in H, we have that x ↦ ⟨π(x)ψ, ϕ⟩ is |μ|-measureable for all μ ∈ M(G) and ψ ∈ H.

A quick calculation now shows that

∫_{G}⟨π(x)π_{A}(ν)φ, ϕ⟩dμ(x) = ∫_{G} ∫_{G} l(xy)dν(y)dμ(x) = ∫_{G} l(z)d(μ ∗ ν)(z)
= ⟨π_{A}(μ ∗ ν)φ, ϕ⟩ = ⟨π_{A}(μ)π_{A}(ν)φ, ϕ⟩.

Since the collection of elements of the form π_{A}(ν)φ is dense in H, the equality

∫_{G}⟨π(x)ψ, ϕ⟩dμ(x) = ⟨π_{A}(μ)ψ, ϕ⟩

holds in general. Since ϕ ∈ H was arbitrary, this proves our theorem.

The theorem above can be applied specifically to the subalgebra M_{a}(G) of G. In doing so, the representation that we end up with is in fact a continuous representation. Before we prove this, we must establish a technical lemma.

Lemma 65. Let h be a real or complex-valued function on G. Then there exists a λ-measureable fuction h̃ on G equal to h locally λ-a.e., and h̃1_{B} is Borel measureable for all σ-compact subsets B ⊂ G.

Proof. We prove this for the case that h is real-valued and positive, and the more general case follows immediately. Let U be a conditionally compact symmetric neighborhood of e in G, and set L = ⋃_{i=1}^{∞} U^{i}^{9}. Then if x, y ∈ L, x ∈ U^{j} and y ∈ U^{k} for some integers j, k > 0. It follows that xy ∈ U^{j+k} ⊂ L, and therefore L is a subgroup of G. The set L is open, since it is a union of open sets. Since L is a subgroup, L^{c} = ⋃_{g∈G\L} gL. Since translation is a continuous operation, gL is open for every g ∈ G. We conclude that L^{c} is also a union of open sets, and therefore L is closed. By its construction, L is σ-compact.

Let S be a subset of G such that {sL}_{s∈S} forms the collection of all distinct left cosets of L in G, and define h_{s} = h1_{sL} for all s ∈ S. For each s ∈ S, sL is σ-compact, and therefore there exists a monotone increasing sequence of compact sets {F_{i}^{(s)}}_{i=1}^{∞} such that sL = ⋃_{i=1}^{∞} F_{i}^{(s)}. For every positive integer n, define g_{n}^{(s)} to be the function supported on F_{n}^{(s)} such that

9An open set U is called conditionally compact if its closure is compact. It is called symmetric if x^{−1} ∈ U for every x ∈ U.

g_{n}^{(s)}(x) = \{ \begin{matrix}h_{s}(x) & h_{s}(x) ⩽ n; \\ n & otherwise \end{matrix} ,

for all x ∈ F_{n}^{(s)}. Then g_{n}^{(s)} ↑ h_{s} and each g_{n}^{(s)} is bounded with compact support, and therefore is equal to a λ-measureable function g̃_{n}^{(s)} λ-a.e.. It follows that g̃_{n}^{(s)} → h̃_{s}, where h̃_{s} is a λ-measureable function and h̃_{s} = h_{s} λ-a.e.. The cosets {sL}_{s∈S} are mutually disjoint, and their union is G, so the function h̃ defined by h̃(x) = h_{s}(x) for x ∈ sL is a well-defined function on G. By its definition, h̃ = h locally λ-a.e.. If B ⊂ G is σ-compact, there exists a countable subcover {s_{i}L}_{i=1}^{∞} of the open cover {sL}_{s∈S} of B, from which it follows h̃1_{B} = 1_{B} ∑_{i=1}^{∞} h_{s_{i}}. This is the product of Borel measureable functions, and therefore Borel measureable. This proves our lemma.

Theorem 66. Let π_{A} be a ∗-representation of the algebra M_{a}(G) by operators on a Hilbert space H such that for all nonzero f ∈ H there exists a μ ∈ M_{a}(G) such that ρ(μ)f ≠ 0. Then there exists a continuous unitary representation π of G by unitary operators on H satisfying Equation (21) for all μ ∈ M_{a}(G) and f, h ∈ H. Moreover, this representation is unique in the sense that it is the only continuous unitary representation of G satisfying Equation (21).

Proof. We know that M_{a}(G) is a subalgebra of M(G) and that every bounded linear functional on M_{a}(G) has the form

μ ↦ ∫_{G} h(x)dμ(x)

for some h ∈ L_{∞}(G). By Lemma (65), we may assume that 1_{S}h is Borel measureable for all σ-compact subsets S ⊂ G. It follows that M_{a}(G) satisfies the assumptions of Theorem (64), so we have a unitary representation π of G satisfying Equation (21) that is weakly μ-measureable for all μ ∈ M(G). We now wish to show that π is continuous. As in the proof of Theorem (63) H can be decomposed into the direct sum H = ⨁_{γ∈Γ} H_{γ} of π_{A}-invariant subspace of H, where the restriction of π_{A} to H_{γ} is cyclic for all γ ∈ Γ. For each γ ∈ Γ, let φ_{γ} be a cyclic vector of the restriction of π_{A} to H_{γ}. Then as in the proof of Theorem (63), for any μ ∈ M_{a}(G)

∥π(x)π_{A}(μ)φ_{γ} − π_{A}(μ)φ_{γ}∥ = ∥π_{A}(δ_{x} ∗ μ)φ_{γ} − π_{A}(μ)φ_{γ}∥

⩽ ∥δ_{x} ∗ μ − μ∥∥φ_{γ}∥ = ∥L(x)\frac{dμ}{dλ} − \frac{dμ}{dλ}∥_{1} ∥φ_{γ}∥.

The mapping L(·)f_{μ} : x ↦ L(x)\frac{dμ}{dλ} is right uniformly continuous, so this shows us that the mapping π(·)ψ : x ↦ π(x)ψ is left uniformly continuous for all ψ ∈ Orb(φ_{γ}, π_{A}). Since Orb(φ_{γ}, π_{A}) is dense in H_{γ}, this proves left uniform continuity for all ψ ∈ H_{γ}. The uniform continuity of π(·)ψ for all ψ ∈ H follows immediately. Thus π is continuous.

If ρ is another continuous representation of G by unitary operators on H satisfying Equation (21), then

0 = ⟨π_{A}(μ)ψ, ϕ⟩ − ⟨π_{A}(μ)ψ, ϕ⟩ = ∫_{G}⟨(π(x) − ρ(x))ψ, ϕ⟩dμ(x)

for all μ ∈ M_{a}(G) and ψ, ϕ ∈ H. In particular, if ⟨(π(x) − ρ(x))ψ, ϕ⟩ ≠ 0 for some x ∈ G, then by continuity we know that there exists a set of nonzero positive λ measure S ⊂ G such that the real or imaginary part of ⟨(π(x) − ρ(x))ψ, ϕ⟩ does not change sign on S. This leads immedately to a contradiction by using dμ = 1_{S}dx in the integral above. This proves our theorem.

2.6. The Regular Representation of M(G) and Gel’fand-Raĭkov

Our most important example of a ∗-representation of M(G) will be the “regular representation”, which is a faithful representation by operators on the Hilbert space L_{2}(G).

Definition 67. The left regular representation L_{A} of M(G) is a representation of M(G) by bounded operators on the Hilbert space L_{2}(G) defined by L_{A}(μ)f = μ ∗ f.

The reason for our notation is that the representation L_{A} of M(G) is closely related to the representation L of G. In fact, a quick calculation shows that L_{A}(δ_{a})f = L(a)f for all a ∈ G. It is the restriction of this representation to the subalgebra M_{a}(G) of M(G) that will be used to prove the existence of a large collection of continuous unitary representations of G. The next theorem takes up the task of showing that the regular representation is in fact a well-defined faithful representation of G.

Theorem 68. The a faithful representation L_{A} of M(G) is a well-defined faithful ∗-representation of M(G) by bounded operators on L_{2}(G).

Proof. For any f ∈ L_{2}(G), we have that ∥L_{A}(μ)f∥ = ∥μ ∗ f∥ ⩽ ∥μ∥∥f∥_{2}. It follows that ∥L_{A}(μ)∥ ⩽ ∥μ∥, and so L_{A}(μ) is a bounded operator on L_{2}(G). Moreover, the linearity of L_{A} follows immediately from the properties of convolution. If f ∈ C_{00}(G), then

L_{A}(μ ∗ ν)f = (μ ∗ ν) ∗ f = μ ∗ (ν ∗ f) = L_{A}(μ)(ν ∗ f) = L_{A}(μ)L_{A}(ν)f.

Since C_{00}(G) is dense in L_{2}(G), it follows that L_{A}(μ ∗ ν)f = L_{A}(μ)L_{A}(ν)f for all f ∈ L_{2}(G). Thus L_{A} is a representation of M(G) by bounded operators.

If f, g ∈ C_{00}(G), then by Fubini’s theorem, we have that

⟨π_{A}(μ^{∗})f, h⟩ = ∫_{G}(μ^{∗} ∗ f)(x)\overline{h(x)}dx = ∫_{G} \overline{h(x)}\overline{∫_{G} \overline{f(yx)}dμ(y)}dx
= \overline{∫_{G} ∫_{G} h(x)\overline{f(yx)}dxdμ(y)} = \overline{∫_{G} ∫_{G} h(y^{−1}x)\overline{f(x)}dxdμ(y)}

= \overline{∫_{G} \overline{f(x)} ∫_{G} h(y^{−1}x)dμ(y)dx} = \overline{∫_{G} \overline{f(x)}(μ ∗ h)(x)dx} = ⟨f, π_{A}(μ)h⟩.

Since C_{00}(G) is dense in L_{2}(G), this shows that ⟨π_{A}(μ^{∗})f, h⟩ = ⟨f, π_{A}(μ)h⟩ for all f, h ∈ L_{2}(G). Thus (π_{A}(μ^{∗}))^{∗} = π_{A}(μ), and we conclude that L_{A} is a ∗-representation of M(G). Lastly, to show that the representation is faithful, let 0 ≠ μ ∈ M(G), let θ be the continuous function θ(x) = x^{−1} and let f ∈ C_{00}(G) be such that ∫_{G} f ∘θdμ ≠ 0. Then f ∘ θ is right uniformly continuous and

|μ ∗ f(y) − μ ∗ f(x)| = |∫_{G} f(z^{−1}y) − f(z^{−1}x)dμ(z)|
= |∫_{G} L(y)f ∘ θ − L(z^{−1})f ∘ θdμ(z)| ⩽ ∥L(y)f ∘ θ − L(z)f ∘ θ∥_{u}∥μ∥,

so that μ ∗ f must also be continuous. Since in particular (μ ∗ f)(e) = ∫_{G} f ∘ θdμ ≠ 0, this allows us to conclude that ∥μ ∗ f∥_{2} ≠ 0 and therefore L_{A}(μ) is not identically zero on L_{2}(G). This proves our theorem.

We are now fully prepared to prove the Gel’fand-Raĭkov theorem, which establishes the fact that the collection of all irreducible, unitary, continuous representations in some sense “separates points”, i.e. that for every x, y ∈ G with x ≠ y there are irreducible, unitary, continuous representations π and π′ with π(x) ≠ π′(y). The result is fundamental and important in the sense that it may be combined later with a version of the Stone-Weierstrass theorem to obtain that the closed linear space of all functions of the form x ↦ ⟨π(x)f, h⟩ are closed in C_{0}(G) (and therefore L_{p}(G)).

Theorem 69 (Gel’fand-Raĭkov). A locally compact group G has sufficiently many irreducible, unitary continuous representations. That is, for every non-identity element g ∈ G, there exists a continuous, irreducible unitary representation π of G with π(g) ≠ I.

Proof. Let U be a symmetric neighborhood of e with λ(U) < ∞ and g ∉ U^{2}. Then aU ∩ U = ∅. Let μ be the measure in M_{a}(G) such that dμ = 1_{U}dλ. Then d(δ_{g} ∗ μ) = L(g)1_{U}dx = 1_{gU}dx and ∥δ_{g} ∗ μ − μ∥ = ∥1_{gU} − 1_{U}∥ = 2λ(U), and therefore δ_{a} ∗ μ ≠ μ. Since L_{A} is faithful, it follows that L_{A}(δ_{g} ∗ μ) ≠ π_{A}(μ). For every g ∈ G, there exists an irreducible ∗-representation π_{A} of M_{a}(G) over a Hilbert space H such that π_{A}(δ_{g} ∗ μ) ≠ π_{A}(μ) (see [2] pp. 330). By Theorem (66), there exists a continuous unitary representation π of G such that

⟨π_{A}(ν)ψ, ϕ⟩ = ∫_{G}⟨π(x)ψ, ϕ⟩dν(x)

for all ν ∈ M_{a}(G) and all ψ, ϕ ∈ H. Furthermore,

∫_{G}⟨π(x)ψ, ϕ⟩dδ_{g} ∗ μ(x) = ∫_{G} ∫_{G}⟨π(yx)ψ, ϕ⟩dδ_{g}(y)dμ(x)
= ∫_{G}⟨π(gx)ψ, ϕ⟩dμ(x) = ∫_{G}⟨π(g)π(x)ψ, ϕ⟩dμ(x)

from which it follows that

∫_{G}⟨π(g)π(x)ψ, ϕ⟩dμ(x) ≠ ∫_{G}⟨π(x)ψ, ϕ⟩dμ(x).

If π is the identity, this is a contradiction, so π ≠ I. Any subspace V ⊂ H invariant with respect to the representation π is also invariant with respect to π_{A} by Theorem (60), and therefore 0 or H, since π_{A} is irreducible. It follows that π is irreducible. This proves our theorem.

Of special importance to us are compact groups, which satisfy the property that all irreducible, unitary, continuous representations must be finite dimensional. The Gel’fand-Raĭkov theorem then tells us that there are sufficiently many representations of this kind. Similarly, for locally compact abelian groups, all the irreducible, unitary representations must be 1-dimensional, and the Gel’fand-Raĭkov theorem tells us we have sufficiently many of these as well.

Theorem 70. Every irreducible continuous representation π of a compact group G by unitary operators on a Hilbert space H is finite-dimensional.

Proof. Recall that for compact groups we use normalized Haar measure (λ(G) = 1). Let G be a compact group and π be an irreducible continuous representation of G by unitary operators on a Hilbert space H. For any ψ, φ ∈ H, the function ⟨π(·)φ, ψ⟩ ∈ L_{2}(G). In fact,

|⟨⟨π(·)φ, ψ⟩, ⟨π(·)φ, ϕ⟩⟩| = |∫_{G}⟨π(x)φ, ψ⟩\overline{⟨π(x)φ, ϕ⟩}| dx
⩽ ∫_{G} ∥π(x)φ∥^{2}∥ψ∥∥ϕ∥dx = ∥φ∥^{2}∥ψ∥∥ϕ∥.

By the properties of the inner product, ⟨⟨π(·)φ, ψ⟩, ⟨π(·)φ, ϕ⟩⟩ therefore a bounded functional that is linear in ϕ and conjugate-linear in ψ for fixed φ, and we can conclude that there exists a bounded operator B_{φ} on H such that

⟨B_{φ}ϕ, ψ⟩ = ⟨⟨π(·)φ, ψ⟩, ⟨π(·)φ, ϕ⟩⟩.

for all ϕ, ψ ∈ H. Furthermore, due to the invariance of the Haar integral with respect to left translation,

⟨B_{φ}π(a)ϕ, ψ⟩ = ⟨⟨π(·)φ, ψ⟩, ⟨π(·)φ, π(a)ϕ⟩⟩ = ∫_{G}⟨π(x)φ, ψ⟩\overline{⟨π(x)φ, π(a)ϕ⟩}dx
= ∫_{G}⟨π(a^{−1})π(x)φ, π(a^{−1})ψ⟩\overline{⟨π(a^{−1})π(x)φ, ϕ⟩}dx
= ∫_{G}⟨π(a^{−1}x)φ, π(a^{−1})ψ⟩\overline{⟨π(a^{−1}x)φ, ϕ⟩}dx
= ∫_{G}⟨π(x)φ, π(a^{−1})ψ⟩\overline{⟨π(x)φ, ϕ⟩}dx

= ⟨⟨π(·)φ, π(a^{−1})ψ⟩, ⟨π(·)φ, ϕ⟩⟩ = ⟨B_{φ}ϕ, π(a^{−1})ψ⟩ = ⟨π(a)B_{φ}ϕ, ψ⟩.

Since ϕ, ψ ∈ H were taken arbitrarily, this shows that B_{φ} commutes with π(a) for all a ∈ G. Since π is irreducible, it follows that B_{φ} = α_{φ}I for some α_{φ} ∈ ℂ. In particular, this shows us that

α_{φ}∥ψ∥^{2} = ⟨B_{φ}ψ, ψ⟩ = ∥⟨π(·)φ, ψ⟩∥_{2}^{2}

for all ψ, φ ∈ H. Let θ be the continuous function on G defined by θ(x) = x^{−1}. Reversing φ and ψ above, then gives us

α_{ψ}∥φ∥^{2} = ∥⟨π(·)ψ, φ⟩∥_{2}^{2} = ∥⟨φ, π(·)ψ⟩∥_{2}^{2} = ∥⟨π(·)φ, ψ⟩ ∘ θ∥_{2}^{2}
= ∥⟨π(·)φ, ψ⟩∥_{2}^{2} = α(φ)∥ψ∥^{2},

where we have used the fact that the Haar integral is invariant under inversion for unimodular groups (in particular, for compact groups). Now if we take ∥φ∥ = 1, and set c = α_{φ} we find that α_{ψ} = c∥ψ∥^{2} for all ψ ∈ H. The function |⟨π(·)φ, φ⟩|^{2} is continuous and is equal to ∥φ∥^{2} = 1 at e. The above then also tells us that 0 < ∥⟨π(·)φ, φ⟩∥_{2}^{2} = α_{φ}∥φ∥^{2} = c, so that necessarily c > 0.

Now if we take any collection of orthonormal vectors {ξ_{i}}_{i=1}^{n} ⊂ H, we have that

∥⟨π(·)ξ_{i}, ξ_{j}⟩∥_{2}^{2} = α_{ξ_{i}}∥ξ_{j}∥^{2} = c∥ξ_{i}∥^{2}∥ξ_{j}∥^{2} = c.

Given x ∈ G, since π(x) is unitary, the collection {π(x)ξ_{i}}_{i=1}^{n} is orthonormal and may be extended to an orthonormal basis B of H. We then have the inequality

∑_{i=1}^{n} |⟨π(x)ξ_{i}, ψ⟩|^{2} ⩽ ∑_{ϕ∈B} |⟨ϕ, ψ⟩|^{2} = ∥ϕ∥^{2},

which holds for any x ∈ G and ϕ ∈ H. In particular, taking ϕ = ξ_{j}, we find

nc = ∑_{i=1}^{n} ∥⟨π(·)ξ_{i}, ξ_{j}⟩∥_{2}^{2} ⩽ ∫_{G} ∑_{i=1}^{n} |⟨π(·)ξ_{i}, ξ_{j}⟩|^{2} ⩽ ∫_{G} ∥ξ_{j}∥^{2} = 1.

In particular, this says that n ⩽ 1/c, so that n must be finite. This proves our theorem.

Corollary 71. Let G be a compact group. Then G has sufficiently many irreducible, unitary continuous representations by unitary matrices.

Proof. This follows immediately from the Gel’fand-Raĭkov theorem and the fact that all irreducible continuous unitary representations of G are finite dimensional.

Definition 72. Let S be a semigroup. A homomorphism of S into ℂ is called a multiplicative function on S. A nontrivial bounded multiplicative function on S is called a semicharacter of S. A semicharacter of a group G is called a character of G.

The assumption that a semicharacter of a semigroup S is bounded requires that it maps S into the disk {λ ∈ ℂ : |λ| ⩽ 1}. Moreover if χ is a character of a group G, then χ is a homomorphism of G into 𝕋, since if for some g ∈ G |χ(g)| < 1, then 1 = |χ(e)| = |χ(g^{−1})χ(g)| = |χ(g^{−1})||χ(g)|, so that χ(g^{−1}) > 1, which is a contradiction. Thus every character of G is a 1-dimensional unitary representation of G over the Hilbert space ℂ and every 1-dimensional unitary representation π of G by operators on a Hilbert space defined by the orthonormal basis {f} defines a character χ via the relation π(g)(f) = χ(g)f. This allows us to identify the collection of all characters of G with the collection of all 1-dimensional unitary representations of G.

Theorem 73. Every irreducible continuous unitary representation π of a locally compact abelian group G is 1-dimensional.

Proof. Since G is abelian, for any a ∈ G, π(a) commutes with π(x) for all x ∈ G. Since π is irreducible, it follows that π(a) = λI for some λ ∈ ℂ. Every subspace of H is invariant under an operator of the form λI, so H must be 1-dimensional.

Corollary 74. Every locally compact abelian group G has sufficiently many continuous characters.

Proof. This is an immediate consequence of the Gel’fand-Raĭkov theorem and the fact that all irreducible, continuous, unitary representations of G are 1-dimensional.

CHAPTER 3. UNITARY REPRESENTATIONS OF COMPACT GROUPS

3.1. Introduction

The purpose of this chapter is to set up the machinery necessary to do harmonic analysis on compact groups, and to prove the famous Peter-Weyl theorem that establishes that the coordinate functions of a compact group G determine a dense orthonormal set. For the most part, the notation from the previous chapter will carry over. However, in this chapter we will always specifically assume that G is compact unless otherwise stated. It follows from Theorem (69) that the irreducible, continuous, unitary operators on G will always be finite-dimensional.

3.2. Basic Definitions and Facts

We begin this chapter as well with a rather large collection of definitions to set the stage.

Definition 75. Let S be a semigroup and X and X′ reflexive Banach spaces. Let π and ρ be two representations of S by bounded operators on X and X′ respectively. A bounded linear transformation T : X → X′ satisfying T π(g) = ρ(g)T for all g ∈ S is called an intertwining operation. The collection of all intertwining operations on G is a closed linear subspace of the Banach space B(X, X′) of all bounded linear transformations from X into X′ and is denoted by Twist(π, ρ).

It is clear that two representations are equivalent if Twist(π, ρ) contains a linear isometry. Moreover, if π = ρ, then Twist(π, ρ) = {αI : α ∈ ℂ}. (see Corollary (92)).

Definition 76. The equivalence of irreducible unitary operators defined above is an equivalence relation, and we call the collection of equivalence classes under this relation the dual object of G and denote it by� . For any given equivalence class σ ∈� , we write π_{σ} for a representation in σ and H_{σ} for its corresponding Hilbert space.

The equivalence class of a continuous character of G will be a singleton set, and so we may identify the subset of� of all 1-dimensional representations with the collection� of all continuous characters on G. Definition 77. For a compact group G and a fixed σ ∈� , it is obvious that all π_{σ} ∈ σ operate on Hilbert spaces of the same finite dimension. We let d_{σ} denote the dimension of these representation spaces. We denote by 1 both the trivial 1dimensional representation of G and its corresponding equivalence class. Like all 1-dimensional representations, its equivalence class consists of itself alone. Given an element σ ∈� with representation π_{σ} ∈ σ and a basis {ξ_{i}}_{i=1}^{d_{σ}} for H_{σ}, we define a collection of functions π_{jk}^{(σ)} : G → ℂ by

π_{jk}^{(σ)}(g) = ⟨π_{σ}(g)ξ_{k}, ξ_{j}⟩

for all g ∈ G. The functions π_{jk}^{(σ)} are called coordinate functions for π_{σ} ∈ σ and the basis {ξ_{1}, …, ξ_{d_{σ}}}. In particular, π_{jk}^{(σ)}(g) is the matrix of the operator π_{σ}(g) in this basis. We note that the mapping g ↦ π_{jk}^{(σ)}(g) is a representation of g by unitary matrices. In particular, we have the relations (i)

π_{σ}(g)ξ_{k} = ∑_{j=1}^{d_{σ}} π_{jk}^{(σ)}(g)ξ_{j}

(ii)

π_{jk}^{(σ)}(gh) = ∑_{l=1}^{d_{σ}} π_{jl}^{(σ)}(g)π_{lk}^{(σ)}(h)

(iii)

π_{jk}^{(σ)}(g^{−1}) = \overline{π_{kj}^{(σ)}(g)}

Definition 78. For σ ∈� and π_{σ} ∈ σ with representation space H_{σ}, we define Trig_{σ}(G) to be the collection of all finite complex linear combinations of functions of the form

⟨π_{σ}(·)ψ, ϕ⟩

with ψ, ϕ ∈ H_{σ}.

The definition of Trig_{σ}(G) is independent of the choice of representation π_{σ}. In fact, if ρ ∈ σ is another representation with representation space H′_{σ}, there exists a linear isometry T : H_{σ} → H′_{σ} such that ρ(g)T = T π_{σ}(g) for all g ∈ G and therefore any function of the above form can be rewritten as

g ↦ ⟨π_{σ}(g)ψ, ϕ⟩ = ⟨T π_{σ}(g)ψ, T ϕ⟩ = ⟨ρ(g)T ψ, T ϕ⟩ = ⟨ρ(g)ψ′, ϕ′⟩

where ψ′ = T ψ and ϕ′ = T ϕ are vectors in H′_{σ}. Conversely, in a similar way any function of the form g ↦ ⟨ρ(g)ψ′, ϕ′⟩ for ψ′, ϕ′ ∈ H′_{σ} is expressable in the form g ↦ ⟨π_{σ}(g)ψ, ϕ⟩ for some ψ, ϕ ∈ H_{σ}.

If 𝔭 ⊂� , then we denote by Trig_{𝔭}(G) the smallest linear space of functions containing ⋃_{σ∈𝔭} Trig_{σ}(G). If 𝔭 =� , we write Trig_{𝔭}(G) as Trig(G). Functions in Trig(G) are called trigonometric polynomials on G.

The justification for calling functions Trig(G) trigonometric polynomials, stems from the special case of G = T. In this case,� ≡� = {g ↦ g^{k} : k ∈ ℤ}. Thus the functions g ↦ ⟨π(g)ξ, η⟩ all have the form g ↦ αg^{n}, for some α ∈ ℂ and n ∈ ℤ. Since each g is of the form g = e^{it}, Trig(T) consists of all functions of the form ∑_{k=−n}^{n} α_{k} exp(ikt), which is the usual collection of trigonometric polynomials.

We now begin to outline some of the properties of trigonometric polynomials.

Fact 79. The function spaces Trig_{σ}(G) and Trig_{𝔭}(G) for σ ∈� and 𝔭 ⊂� are invariant under left and right translation. In fact, for any representation π of G by unitary operators on a Hilbert space H

(R(a)⟨π(·)ψ, ϕ⟩)(g) = ⟨π(ga)ψ, ϕ⟩ = ⟨π(g)(π(a)ψ), ϕ⟩;
(L(a)⟨π(·)ψ, ϕ⟩)(g) = ⟨π(ag)ψ, ϕ⟩ = ⟨π(g)ψ, π(a^{−1})ϕ⟩.

3.3. Trigonometric Polynomials and Stone-Weierstrass

Since representations π_{σ} ∈ σ ∈� are continuous, they are weakly continuous and therefore Trig_{𝔭}(G) ⊂ C(G) = C_{00}(G). Moreover, as long as 𝔭 is closed under certain operations Trig_{𝔭}(G) will actually be an algebra of functions. This is a very helpful property, since it will allow us to apply the Stone-Weierstrass theorem and obtain the property that Trig_{𝔭}(G) is dense in C_{00}(G). In this section, we establish the requirements necessary for Trig_{𝔭}(G) to be an algebra, and in particular show that Trig(G) is dense in C_{00}(G). We begin with some definitions.

Definition 80. Let H be a Hilbert space of dimension d, and let {ξ_{i}}_{i=1}^{d} be an orthonormal basis of H. Define a mapping D : H → H by

D (∑_{i=1}^{d}⟨ψ, ξ_{i}⟩ξ_{i}) = ∑_{i=1}^{d} \overline{⟨ψ, ξ_{i}⟩}ξ_{i}.  (23)

A transformation of the form D above is called a conjugation transformation.

The properties of conjugate transformations are outlined in the next lemma.

Lemma 81. Let D be a conjugate transformation of a d-dimensional Hilbert space H. Then D is a continuous conjugate-linear bijection on H with D^{2} = I. Let T : H → H′ be a linear operator from H to another Hilbert space H′. Then DT D is also a linear operator. If T is unitary, then DT D is unitary. If T is continuous, so too is DT D. If H = H′ and V is a T -invariant subspace of H, then DV is DT D-invariant. If T is a linear isometry, then DT is a conjugate transformation of H. If D′ is another conjugate transformation of H, then there exists a linear isometry L on H such that D′ = DL.

Proof. Let D be a conjugate transformation of H and let {ξ_{i}}_{i=1}^{d} be an orthonormal basis for H satisfying Equation (23). The fact that D is conjugate-linear and bijective follows immediately from its definition. The fact that DT D is linear and D^{2} = I follows immediately from the definition and an obvious calculation, since elements are conjugated twice. Moreover

∥Dψ∥^{2} = ∥∑_{i=1}^{d} \overline{⟨ψ, ξ_{i}⟩}ξ_{i}∥^{2} ⩽ ∑_{i=1}^{d} |\overline{⟨ψ, ξ_{i}⟩}|^{2} = ∥ψ∥,

so D is norm-preserving, and therefore continuous. In particular, since D is normpreserving, DT D is unitary if T is. The composition of continuous mappings is continuous, so DT D is continuous when T is. If H = H′ and V is a subspace of H invariant under T, then any Dψ ∈ DV satisfies DT D(Dψ) = DT ψ ∈ DV. It follows that DV is a DT D-invariant subspace of H. The last two properties of the theorem above follow simply from the properties of change of base in a Hilbert space.

From the above lemma, we immediately obtain the following theorem.

Theorem 82. Let π be a continuous, irreducible, unitary representation of G with representation space H, and let D be a conjugate transformation on H. Then DπD : x ↦ Dπ(x)D is also a continuous, irreducible, unitary representation of G. If D′ is another conjugate transformation of H and ρ is equivalent to π, then D′ρD′ is equivalent to DπD. In particular, D′πD′ is equivalent to DπD.

Proof. From the previous lemma, we know that D = D^{−1}. If x, y ∈ G, then Dπ(xy)D = Dπ(x)π(y)D = Dπ(x)DDπ(y)D, and so DπD is a representation of G. From the above, we also know that DπD is continuous. The lemma above also shows that for any x ∈ G, any Dπ(x)D-invariant subspace of V ⊂ H corresponds to a π(x) invariant subspace DV, and since D is a bijection, this allows us to conclude that DπD is irreducible. The previous lemma also shows that DπD is unitary if π is. If D′ is another conjugate transformation of H and ρ is equivalent to π with corresponding representation space H′, then there exists isometries T : H → H′ and L : H → H such that T π(g) = ρ(g)T for all g ∈ G and DL = D′. Since D = D^{−1} and D′ = (D′)^{−1}, we also have that L^{−1}D = D′. It follows that

D′ρ(g)D′ = DLρ(g)L^{−1}D = DLT π(g)T^{−1}L^{−1}D = DLT D(Dπ(g)D)(DLT D)^{−1}.

By the previous lemma, DLT D is an isometry from H to H′. This proves our theorem.

The theorem we just proved allows us to justify the following definition, which defines a conjugation operation on� .

Definition 83. Let π be a representation of G by unitary operators on a Hilbert space H. A representation of the form DπD : x ↦ Dπ(x)D called a conjugate representation of π. Let σ ∈� . The conjugate of the dual element \overline{σ} is the member of � given by the equivalence class of all representations conjugate to a representation π_{σ} ∈ σ. A subset 𝔭 ⊂� is closed under conjugation if \overline{σ} ∈ 𝔭 for all σ ∈ 𝔭.

The importance of the conjugates of dual elements, follows from the fact that if 𝔭 ⊂� is closed under the conjugation operation defined above, then Trig_{𝔭}(G) is closed under conjugation as well. To show this, take any π_{σ} ∈ σ ∈� with corresponding representation space H_{σ}. Then fix a basis {ξ_{i}}_{i=1}^{d_{σ}} of H_{σ}, let π_{ij}^{(σ)} denote the usual coordinate functions with respect to this basis, and let D be the conjugation operation on H with respect to this basis. Then Dξ_{i} = ξ_{i} for all i and

⟨Dπ(g)Dξ_{i}, ξ_{j}⟩ = \overline{⟨DDπ(g)Dξ_{i}, Dξ_{j}⟩} = \overline{⟨π(g)ξ_{i}, ξ_{j}⟩} = \overline{π_{ij}^{(σ)}}.

Thus the coordinate functions of DπD with respect to the basis {ξ_{i}}_{i=1}^{d_{σ}} are the complex conjugates of those of π.

Now to establish Trig_{𝔭}(G) as an algebra, it remains only to show that it is closed under products. This requires that we have 𝔭 closed under an additional operation, namely tensor product. We have the following definition.

Definition 84. Let π_{1} and π_{2} be two irreducible, continuous, unitary representations G with corresponding Hilbert spaces H_{1} and H_{2}, respectively. The tensor product of the representations π_{1} ⊗ π_{2} is the representation of G defined by π_{1} ⊗ π_{2}(g) = π_{1}(g) ⊗ π_{2}(g) that acts on the Hilbert space H_{1} ⊗ H_{2}^{10}.

The tensor product has many useful properties, which we do not prove here. The properties that we will need are that the tensor product of two continuous or unitary operators has the same property. In particular, the tensor product of two continuous, unitary representations is continuous and unitary. It is not true, however, that the tensor product of two irreducible representations is irreducible. Instead, we must consider the decomposition of the Hilbert space H_{1} ⊗ H_{2} into π_{1} × π_{2}-invariant subspaces. If H_{1} has dimension d_{1} and H_{2} has dimension d_{2}, then the dimension of H_{1} ⊗ H_{2} is d_{1}d_{2}, and therefore finite. Thus there are no problems with simply taking a decomposition

H_{1} ⊗ H_{2} = ∑_{i=1}^{n} m_{i}H′_{i},  (24)

where H′_{i} is a π_{1} × π_{2}-invariant subspace of H_{1} ⊗ H_{2} for all i and we use the notation mH to denote the direct sum H ⊕H ⊕… H of m copies of a Hilbert space H. Defining π′_{i} to be the restriction of π_{1} ⊗ π_{2} to H′_{i}, this decomposes the representation π into the direct sum of representations

10The tensor product of two Hilbert spaces is always over the field ℂ.

π_{1} ⊗ π_{2} = ∑_{i=1}^{n} m_{i}π′_{i},  (25)

where we have used the notation mπ to denote the direct sum π ⊕ π ⊕ … π of m copies of the representation π. If the representations π_{1} and π_{2} (and therefore π_{1} ⊗π_{2}) are continuous and unitary, then the restrictions π′_{i} surely are, too. Moreover n is finite and m_{i} is finite for each i, since the dimension of H_{1} ⊗ H_{2} is finite. We have the following definition. Definition 85. A subset 𝔭 ⊂� is closed under tensor product if given any σ_{i} ∈� and π_{i} ∈ σ_{i} with corresponding representation space H_{i} for i = 1, 2, the equivalence classes of the elements π′_{i} of the decomposition given in Equation (24) and Equation

(25) are contained in 𝔭.

In particular� is closed under tensor products by definition. If 𝔭 is closed under tensor product, then take any σ_{i} ∈� and π_{i} ∈ σ_{i} with corresponding representation space H_{i} for i = 1, 2. For any ψ_{i}, ϕ_{i} ∈ H_{i}, we have that

⟨(π_{1} ⊗ π_{2})(ψ_{1} ⊗ ψ_{2}), ϕ_{1} ⊗ ϕ_{2}⟩ = ⟨π_{1}ψ_{1} ⊗ πψ_{2}, ϕ_{1} ⊗ ϕ_{2}⟩ = ⟨π_{1}ψ_{1}, ϕ_{1}⟩⟨π_{2}ψ_{2}, ϕ_{2}⟩

by definition. Closure under tensor products means that ⟨(π_{1} ⊗ π_{2})(ψ_{1} ⊗ ψ_{2}), ϕ_{1} ⊗ ϕ_{2}⟩ can be written as a linear combination of members of Trig_{𝔭}(G), and therefore Trig_{𝔭}(G) is closed under products. It follows that if 𝔭 is closed under tensor products and conjugation that Trig_{𝔭}(G) is an algebra.

Recall the Stone-Weierstrass theorem. Theorem 86 (Stone-Weierstrass). Let S be a compact Hausdorff space and C(S) be the algebra of all real continuous functions on S. Let A be a closed subalgebra of C(S) containing the identity. Then A = C(S) if and only if A separates the points of S.

A proof of this theorem will not be given here, but can be found many places. See, for example, [6] pp. 272. We use the following definition.

Definition 87. A subset 𝔭 ⊂� separates points in G if for all x ∈ G with x ≠ e there exists σ ∈ 𝔭 such that π_{σ}(x) ≠ I for some π_{σ} ∈ σ.

This is equivalent to the usual definition of “separates points”, in that x ≠ y if and only if xy^{−1} ≠ e, which is true if and only if π_{σ}(xy^{−1}) ≠ I for some π_{σ} ∈ σ ∈ 𝔭.

The consequence of this theorem is immediate: since the Gel’fand-Raĭkov theorem tells us that members of equivalence classes of� distinguish points in the compact Hausdorff space G, the Stone-Weierstrass theorem tells us the closure of Trig_{𝔭}(G) is C(G). Specifically, we have the following theorem.

Theorem 88. The trigonometric polynomials of G are dense in C(G).

Proof. We have that� separates points by the Gel’fand-Raĭkov theorem. Our theorem then follows from the Stone-Weierstrass theorem and the fact that Trig(G) is an algebra.

An interesting question might be whether or not there is a smaller subcollection 𝔭 of� will be closed under tensor product and conjugation and also separates points in G so that Trig_{𝔭}(G) is dense in C(G). The answer turns out to be no, though we do not yet have the tools to prove it. These are developed by the orthogonality relations, taken up in the next section.

3.4. Orthogonality Relations

In this section we establish the orthogonality relations between the coordinate functions.

Theorem 89. Let π_{1} and π_{1} be unitary representations of an arbitrary group G with representation spaces H_{1} and H_{2} respectively. Suppose that π_{1} is irreducible and that there is a bounded linear isomorphism T ∈ Twist(π_{1}, π_{2}). Then there exists a positive real number β such that βT is a linear isometry from H_{1} to H_{2} and π_{1} ∼ π_{2}.

Proof. Since T ∈ Twist(π_{1}, π_{2}), for every g ∈ G the following diagram commutes.

\xymatrix{H_{1} \ar[r]^{π_{1}(g)} \ar[d]_{T} & H_{1} \ar[d]^{T} & or equivalently & H_{2} \ar[r]^{π_{2}(g)} \ar[d]_{T^{−1}} & H_{2} \ar[d]^{T^{−1}} \\ H_{2} \ar[r]^{π_{2}(g)} & H_{2} &  & H_{1} \ar[r]^{π_{1}(g)} & H_{1}.}

If W is a π_{2}-invariant subspace, then π_{2}(g)(W) ⊂ W for all g ∈ G and therefore π_{1}(g)(T^{−1}(W)) = T^{−1}(π_{2}(g)(W)) ⊂ T^{−1}(W). Thus T^{−1}(W) is π_{1}-invariant and must be 0 or H_{1}, since π_{1} is irreducible. Therefore W must be 0 or H_{2}. We conclude that π_{2} is irreducible.

Define T^{∗} : H_{2} → H_{1} by

⟨T^{∗}ψ_{2}, ψ_{1}⟩_{1} = ⟨ψ_{2}, T ψ_{1}⟩_{2},

for all ψ_{1} ∈ H_{1}, ψ_{2} ∈ H_{2}. This mapping is well-defined: For fixed ψ_{2}, the right side of the above equation is a conjugate-linear (λx ↦ \overline{λ}T^{∗}x, λ ∈ ℂ) and continuous functional on H_{1} and therefore has the form ⟨ϕ, ψ_{1}⟩_{1} for some unique ϕ ∈ H_{1}. We take ϕ = T^{∗}ψ_{2}. It is routine to show the mapping T^{∗} is a bounded linear isomorphism from H_{2} to H_{1}.

For g ∈ G, we also notice that for any ψ_{1} ∈ H_{1} and ψ_{2} ∈ H_{2},

⟨π_{1}(g)T^{∗}ψ_{2}, ψ_{1}⟩_{1} = ⟨T^{∗}ψ_{2}, π_{1}(g^{−1})ψ_{1}⟩_{1} = ⟨ψ_{2}, T π_{1}(g^{−1})ψ_{1}⟩_{2}
= ⟨ψ_{2}, π_{2}(g^{−1})T ψ_{1}⟩_{2} = ⟨π_{2}(g)ψ_{2}, T ψ_{1}⟩_{2} = ⟨T^{∗}π_{2}(g)ψ_{2}, ψ_{1}⟩_{1},

so that π_{1}(g)T^{∗} = T^{∗}π_{2}(g). Since g was arbitrary, this tells us that the adjoint T^{∗} ∈ Twist(π_{2}, π_{1}). Now for any g ∈ G, T^{∗}T π_{1}(g) = T^{∗}π_{2}(g)T = π_{1}(g)T^{∗}T, so that T^{∗}T ∈ Twist(π_{1}, π_{1}) = {αI : α ∈ ℂ}. Thus T^{∗}T = λI for some λ ∈ ℂ. Since T^{∗}T ≠ 0, λ ≠ 0 and for any ψ_{1}, ϕ ∈ H_{1},

λ⟨ϕ, ψ_{1}⟩_{1} = ⟨T^{∗}T ϕ, ψ_{1}⟩_{1} = ⟨T ϕ, T ψ_{1}⟩_{2}.

If ϕ = ψ_{2} ≠ 0, the above tells us λ = ∥T ψ_{1}∥_{2}^{2}/∥ψ_{1}∥_{1}^{2} > 0. Thus taking β = 1/\sqrt{λ} proves the theorem. Theorem 90. Let π_{1} and π_{2} be continuous unitary representations of G with corresponding representation spaces H_{1} and H_{2} and let B ∈ B(H_{1}, H_{2}). Then the mapping A_{B} : H_{1} → H_{2} defined by

⟨A_{B}ψ, ϕ⟩_{2} = C_{B}(ψ, ϕ) = ∫_{G}⟨π_{2}(g^{−1})Bπ_{1}(g)ψ, ϕ⟩_{2}dg  (26)

(where the integral is taken with respect to the normalized Haar measure) is a bounded linear transformation from H_{1} into H_{2}. Moreover A_{B} ∈ Twist(π_{1}, π_{2}). Proof. Let ψ ∈ H_{1} and ϕ ∈ H_{2}. We first show that ⟨π_{2}(g^{−1})Bπ_{1}(g)ψ, ϕ⟩_{2} is a continuous function on G. To show this, write

|⟨π_{2}(x^{−1})Bπ_{1}(x)ψ, ϕ⟩_{2} − ⟨π_{2}(y^{−1})Bπ_{1}(y)ψ, ϕ⟩_{2}|
⩽|⟨Bπ_{1}(x)ψ, (π_{2}(x) − π_{2}(y))ϕ⟩_{2}| + |⟨B(π_{1}(x) − π_{1}(y))ψ, π_{2}(y)ϕ⟩_{2}|
⩽∥B∥ · ∥ψ∥_{1} · ∥π_{2}(x)ϕ − π_{2}(y)ϕ∥ + ∥B∥ · ∥π_{1}(x)ψ − π_{1}(y)ψ∥ · ∥ϕ∥_{2}
⩽∥B∥ · ∥ψ∥_{1} · ∥π_{2}(x) − π_{2}(y)∥ · ∥ϕ∥_{2} + ∥B∥ · ∥π_{1}(x) − π_{1}(y)∥ · ∥ψ∥_{1} · ∥ϕ∥_{2}.

Since π_{1} and π_{2} are continuous, this shows ⟨π_{2}(g^{−1})Bπ_{1}(g)ψ, ϕ⟩_{2} to be continuous. It follows that the integral in Equation (26) is well-defined. Clearly C is a conjugatelinear functional on H_{1} ⊕ H_{2}. Moreover, for fixed ψ ∈ H_{1} C(ψ, ·) is a continuous conjugate-linear functional on H_{2}. Therefore there is a unique φ ∈ H_{2} such that ⟨φ, v⟩_{2} = C(ψ, v) for all ϕ ∈ H_{2}, and we define A_{B}ψ = φ. Clearly A_{B}ψ is a linear transformation from H_{1} into H_{2}. Furthermore, A_{B} is bounded, since

∥A_{B}ψ∥_{2}^{2} = ⟨A_{B}ψ, A_{B}ψ⟩_{2} = C(ψ, A_{B}ψ) = ∫_{G}⟨π_{2}(g^{−1})Bπ_{1}(g)ψ, A_{B}ψ⟩_{2}dg

⩽ sup_{g∈G}{∥Bπ_{1}(g)ψ∥_{2} · ∥A_{B}ψ∥_{2}} ⩽ ∥B∥ · ∥ψ∥_{1} · ∥A_{B}ψ∥_{2}.

In particular, this shows ∥A_{B}∥ ⩽ ∥B∥. Now if we note that

⟨A_{B}π_{1}(h)ψ, ϕ⟩_{2} = ∫_{G}⟨π_{2}(g^{−1})Bπ_{1}(g)π_{1}(h)ψ, ϕ⟩_{2}dg = ∫_{G}⟨π_{2}(g^{−1})Bπ_{1}(gh)ψ, ϕ⟩_{2}dg
= ∫_{G}⟨π_{2}((gh)^{−1})Bπ_{1}(gh)ψ, π_{2}(h^{−1})ϕ⟩_{2}dg
= ∫_{G}⟨π_{2}(g^{−1})Bπ_{1}(g)ψ, π_{2}(h^{−1})ϕ⟩_{2}dg
= ⟨A_{B}ψ, π_{2}(h^{−1})ϕ⟩_{2} = ⟨π_{2}(h)A_{B}ψ, ϕ⟩.

Since ψ ∈ H_{1} and ϕ ∈ H_{2} are arbitrary, this shows that A_{B}π_{1}(h) = π_{2}(h)A_{B} for all h ∈ G. Thus A_{B} ∈ Twist(π_{1}, π_{2}).

Lemma 91 (Schur’s Lemma). Let {S_{α}}_{α∈Ω} and {T_{α}}_{α∈Λ} be irreducible sets of linear operators on vector spaces U and V respectively over a field k, and suppose that there is a linear transformation f : U → V such that for all α ∈ Λ there exists a β ∈ Ω such that the following diagram commutes

\xymatrix{U \ar[r]^{S_{β}} \ar[d]_{f} & U \ar[d]^{f} \\ V \ar[r]^{T_{α}} & V,}

and vise-versa. Then either f = 0 or f is bijective.

Proof. Consider f(U). This is clearly a subspace of V and either f(U) is 0 or V or there exists α ∈ Λ such that T_{α}(f(U)) ⊈ f(U). In the third case, there is a β ∈ Ω such that f(S_{β}(U)) = T_{a}(f(U)), and therefore f(S_{β}(U)) ⊈ f(U). Since S_{β}(U) ⊂ U, this is a contradiction and we must have one of the first two cases. In the first case, Ker f = U. In the second case, we note that

fS_{β}(Ker f) = T_{α}f(Ker f) = T_{α}({0}) = 0

so that in particular S_{β}(Ker f) ⊂ Ker f for all β ∈ Ω. Since the collection {S_{α}}_{α∈Ω} is irreducible and Ker f ≠ U, we conclude that Ker f = 0. Thus either f = 0 or f is a bijection.

Corollary 92. Let {S_{α}}_{α∈Ω} be an irreducible set of linear operators on a vector spaces U over an algebraically closed field k, and suppose that there is a linear transformation f : U → U such that fS_{α} = S_{α}f for all α ∈ Ω. Then f = λI for some λ ∈ k^{11}.

Proof. The function x ↦ det(f − xI) is a polynomial in x over the field k. Since k is assumed to be algebraically closed, there is a root λ ∈ k. Clearly (f − λI)S_{α} = S_{α}(f −λI) for all α ∈ Ω. By Schur’s lemma, f −λI is either a bijection or 0. However, since det(f −λI) = 0, we know that it cannot be a bijection. We conclude f = λI.

Corollary 93. Let π be an irreducible continuous unitary representations of G with corresponding representation spaces U. Then Twist(π, π) = {αI}_{α∈ℂ}.

Proof. Since ℂ is algebraically closed, this follows immediately from Equation (92).

Theorem 94. Let π and ρ be irreducible continuous unitary representations of G with corresponding representation spaces U and V. If there is a B ∈ L(U, V) with A_{B} ≠ 0, then π ∼ ρ.

11A field k is called algebraically closed if every polynomial of degree at least 1 with coefficents in k has a root in k.

Proof. If A_{B} ≠ 0, Schur’s lemma tells us that A_{B} is a bijection. Then Equation (89) tells us A_{B} can be made into an isometry by multiplication by some β ∈ ℂ and π ∼ ρ.

Theorem 95. Let σ_{1} and σ_{2} be distinct elements in� , and let f_{i} ∈ be a function in Trig_{σ_{i}}(G) (i = 1, 2). Then

(i) ∫_{G} f_{1}(g)\overline{f}_{2}(g)dg = 0.

Proof. Let U_{1} and U_{2} be the representation spaces for fixed representations π_{1} ∈ σ_{1} and π_{2} ∈ σ_{2}, respectively. Also let d_{i} denote the dimension of the space U_{i} and let {ξ_{ij}}_{j=1}^{d_{i}} be a fixed orthonormal basis for U_{i}. To prove this theorem, we need only show that

∫_{G}⟨π_{1}(g)ξ_{1i}, ξ_{1j}⟩_{1}\overline{⟨π_{2}(g)ξ_{2k}ξ_{2l}⟩_{2}}dg = 0

for all 1 ⩽ i, j ⩽ d_{1} and 1 ⩽ k, l ⩽ d_{2}. The theorem then follows from the fact that linear combinations of functions of the form ⟨π_{1}(g)ξ_{ij}, ξ_{ik}⟩_{1} make up Trig_{σ_{i}}(G). Given 1 ⩽ i, j ⩽ d_{1} and 1 ⩽ k, l ⩽ d_{2}, we may define a linear transformation B : U_{1} → U_{2} by B(ξ_{1j}) = ξ_{2l} and B(ξ_{1m}) = 0 for m ≠ j. Then we have that

⟨Bπ_{1}(g)ξ_{1i}, π_{2}(g)ξ_{2k}⟩_{2} = ⟨⟨π_{1}(g)ξ_{1i}, ξ_{1j}⟩_{1}ξ_{2l}, ⟨π_{2}(g)ξ_{2k}, ξ_{2l}⟩_{2}ξ_{2l}⟩_{2}
= ⟨π_{1}(g)ξ_{1i}, ξ_{1j}⟩_{1}\overline{⟨π_{2}(g)ξ_{2k}, ξ_{2l}⟩_{2}}.

And therefore

∫_{G}⟨π_{1}(g)ξ_{1i}, ξ_{1j}⟩_{1}\overline{⟨π_{2}(g)ξ_{2k}ξ_{2l}⟩_{2}}dx = ∫_{G}⟨Bπ_{1}(g)ξ_{1i}, π_{2}(g)ξ_{2k}⟩_{2}dg = ⟨A_{B}ξ_{1i}, ξ_{2k}⟩.

By Theorem (94), ⟨A_{B}ξ_{1i}, ξ_{2k}⟩ = 0, which proves our theorem.

Lemma 96. Let π be a continuous irreducible unitary representation of G with ddimensional representation space U. Let B be any linear operator on U. Then the operator A_{B} is equal to d^{−1}tr(B)I.

Proof. The operator A_{B} ∈ Twist(π, π), and therefore is equal to αI for some α ∈ ℂ. Let {ξ_{i}}_{i=1}^{d} be an orthonormal basis in U. Then

tr(B) = tr(π(g^{−1})Bπ(g)) = ∑_{i=1}^{d}⟨π(g)^{−1}Bπ(g)ξ_{i}, ξ_{i}⟩ = ∑_{i=1}^{d}⟨Bπ(g)ξ_{i}, π(g)ξ_{i}⟩,

so that

tr(B) = ∫_{G} tr(B)dg = ∑_{i=1}^{d} ∫_{G}⟨Bπ(g)ξ_{i}, π(g)ξ_{i}⟩dg = ∑_{i=1}^{d} A_{B}(ξ_{i}, ξ_{i}) = tr(A_{B}) = αd

and therefore α = d^{−1}tr(B).

Theorem 97. Let π be a continuous irreducible unitary representation of G with d-dimensional representation space U. Let {ξ_{i}}_{i=1}^{d} be an orthonormal basis in U. Then

∫_{G}⟨π(g)ξ_{i}, ξ_{j}⟩\overline{⟨π(g)ξ_{k}, ξ_{l}⟩}dg = d^{−1}δ_{ik}δ_{jl}.

Proof. Define a linear operator B_{jl} on U by B_{jl}(ξ_{j}) = ξ_{l} and B_{jl}(ξ_{m}) = 0 for m ≠ j. Then A_{B_{jl}} = d^{−1}tr(B_{jl})I = d^{−1}δ_{jl}I and

∫_{G}⟨π(g)ξ_{i}, ξ_{j}⟩\overline{⟨π(g)ξ_{k}, ξ_{l}⟩}dg = ⟨A_{B_{jl}}ξ_{i}, ξ_{k}⟩ = d^{−1}δ_{jl}⟨ξ_{i}, ξ_{k}⟩ = d^{−1}δ_{ik}δ_{jl}.

Corollary 98. Let π_{1} and π_{2} be equivalent continuous irreducible unitary representations of G with corresponding d_{i}-dimensional representation spaces U_{i}, and let T ∈ Twist(π_{1}, π_{2}) be a linear isometry. Let {ξ_{1j}}_{i=j}^{d_{1}} be an orthonormal basis for U_{1} and let ξ_{2i} = T ξ_{1i}. Then

∫_{G}⟨π_{1}(g)ξ_{1i}, ξ_{ij}⟩_{1}\overline{⟨π_{2}(g)ξ_{2k}, ξ_{2l}⟩_{2}}dg = d^{−1}δ_{ik}δ_{jl}.

Proof. A quick calculation shows that

⟨π_{2}(g)ξ_{2k}, ξ_{2l}⟩_{2} = ⟨π_{2}(g)T ξ_{1k}, T ξ_{1l}⟩_{2} = ⟨T π_{1}(g)ξ_{1k}, T ξ_{1l}⟩_{1} = ⟨π_{1}(g)ξ_{1k}, ξ_{1l}⟩_{1}.

Therefore the result follows from the previous theorem.

Gathering up all of our results produces the following orthogonality relations for the coordinate functions

Theorem 99. For each σ ∈� , let {π_{jk}^{(σ)}}_{j,k=1}^{d_{σ}} be a set of coordinate functions for fixed π_{σ} ∈ σ and a fixed basis {ξ_{i}}_{i=1}^{d_{σ}} in the representation space U_{σ} of π_{σ}. Then the set of all functions \sqrt{d_{σ}}π_{jk}^{(σ)} is an orthonormal set in L_{2}(G). In particular, we have

∫_{G} π_{ij}^{(σ)}(g)\overline{π_{kl}^{(σ)}}(g)dg = d_{σ}^{−1}δ_{ik}δ_{jl}.

Some immediate consequences of this theorem are as follows.

Theorem 100. Let σ, ∈� , let {π_{ij}^{(σ)}}_{i,j=1}^{d_{σ}} be coordinate functions defined in the usual way for a fixed basis {ξ_{i}}_{i=1}^{d_{σ}} of the representation space U_{σ} of the representation π_{σ} ∈ σ, and let μ ∈ M(G).

(i)

μ ∗ π_{ij}^{(σ)} = ∑_{k=1}^{d_{σ}} π_{kj}^{(σ)} ∫_{G} \overline{π_{ki}^{(σ)}(g)}dμ(g).

(ii)

π_{ij}^{(σ)} ∗ μ = ∑_{k=1}^{d_{σ}} π_{ik}^{(σ)} ∫_{G} \overline{π_{jk}^{(σ)}(g)}dμ(g).

(iii) If σ′ ∈� and the coordinate functions {π_{ij}^{(σ′)}}_{i,j=1}^{d_{σ}} are defined in the usual way

for a fixed basis,

π_{ij}^{(σ)} ∗ π_{kl}^{(σ′)} = d_{σ}^{−1}δ_{σσ′}δ_{jk}π_{il}^{(σ)},

where δ_{σσ′} = 1 if σ is equivalent to σ′ and 0 otherwise. (iv) We have the adjointness relation (π_{ij}^{(σ)})^{∗} = π_{ji}^{(σ)}. Proof. From the definition of convolution

(μ ∗ π_{ij}^{(σ)})(x) = ∫_{G} π_{ij}^{(σ)}(y^{−1}x)dμ(y) = ∑_{k=1}^{d_{σ}} ∫_{G} π_{ik}^{(σ)}(y^{−1})π_{kj}^{(σ)}(x)dμ(y)

= ∑_{k=1}^{d_{σ}} π_{kj}^{(σ)}(x) ∫_{G} \overline{π_{ki}^{(σ)}(y)}dμ(y).

Similarly

(π_{ij}^{(σ)} ∗ μ)(x) = ∫_{G} π_{ij}^{(σ)}(y)dμ(y^{−1}x) = ∫_{G} π_{ij}^{(σ)}(xy^{−1})dμ(y)

= ∑_{k=1}^{d_{σ}} ∫_{G} π_{ik}^{(σ)}(x)π_{kj}^{(σ)}(y^{−1})dμ(y) = ∑_{k=1}^{d_{σ}} π_{ik}^{(σ)}(x) ∫_{G} \overline{π_{jk}^{(σ)}(y)}dμ(y).

From the second identity applied to dμ = π_{kl}dx, we obtain from the orthogonality relations

(π_{ij}^{(σ)} ∗ π_{kl}^{(σ)})(x) = ∑_{m=1}^{d_{σ}} π_{im}^{(σ)}(x) ∫_{G} \overline{π_{jm}^{(σ)}(y)}π_{kl}^{(σ)}(y)dy = \frac{1}{d_{σ}} ∑_{m=1}^{d_{σ}} π_{im}^{(σ)}(x)δ_{jk}δ_{ml}

= \frac{1}{d_{σ}} π_{il}^{(σ)}(x)δ_{jk}.

From this (iii) is obvious. Lastly, we have that (π_{ij}^{(σ)})^{∗}(g) = \overline{π_{ij}^{(σ)}(g^{−1})} = \overline{⟨π_{σ}(g^{−1})ξ_{i}, ξ_{j}⟩} = \overline{⟨ξ_{i}, π_{σ}(g)ξ_{j}⟩} = ⟨π_{σ}(g)ξ_{j}, ξ_{i}⟩ = π_{ji}^{(σ)}(g).

We now have the toolkit (or utility belt, if the reader prefers) to address the question left over from the previous section. That is, we may show that for Trig_{𝔭}(G) to be dense in C(G), we must have that 𝔭 =� . This is established by the next theorem.

Theorem 101. Let 𝔭 ⊂� be closed under tensor products and conjugation. Then the following are equivalent:

(i) 𝔭 =� ;

(ii) Trig_{𝔭}(G) is dense in C(G);

(iii) 𝔭 separates points in G.

Proof. The proof of (i) implies (iii) and (iii) implies (ii) were already taken care of in the previous section. We need now only show that (ii) implies (i). Suppose that 𝔭 ≠� . Then we may choose σ ∉ 𝔭 with σ ∈� . Let f ∈ Trig_{σ}(G) with ∥f∥_{2}^{2} ≠ 1. If Trig_{𝔭}(G) is dense in C(G), there exists a g ∈ Trig_{𝔭}(G) with ∥f −g∥_{2}^{2} < ∥f∥_{2}^{2}. However, since σ ∉ 𝔭 we know that f and g are orthogonal and therefore ∥f − g∥_{2}^{2} = ∥f∥_{2}^{2} + ∥g∥_{2}^{2} ⩾ ∥f∥_{2}^{2}. This is a contradiction, and we conclude that (ii) implies (i).

The orthogonality results and density results we have thus far attained are summarized in the Peter-Weyl theorem.

Theorem 102 (Peter-Weyl). For each σ ∈� , fix a representation π_{σ} ∈ σ with representation space H_{σ} of dimension d_{σ}. Also fix a basis {ξ_{i}^{(σ)}}_{i=1}^{d_{σ}} of H_{σ} and define the coordinate functions π_{ij}^{(σ)}(x) = ⟨π_{σ}(x)ξ_{i}^{(σ)}, ξ_{j}^{(σ)}⟩ as usual. The collection of functions {d_{σ}^{1/2}π_{ij}^{(σ)} : σ ∈� , 1 ⩽ i, j ⩽ d_{σ}} forms an orthonormal basis for L_{2}(G).

Proof. The fact that the collection is orthonormal follows immediately from the orthogonality relations. The fac that the collection is dense in L_{2}(G) follows from the fact that it is dense in C(G) and C(G) is dense in L_{2}(G).

3.5. The Fourier-Stieltjes Transform

For this entire section, for each σ ∈� , fix a representation π_{σ} ∈ σ with representation space H_{σ} of dimension d_{σ}. Also fix a basis {ξ_{i}^{(σ)}}_{i=1}^{d_{σ}} of H_{σ} and define the coordinate functions π_{ij}^{(σ)}(x) = ⟨π_{σ}(x)ξ_{i}^{(σ)}, ξ_{j}^{(σ)}⟩ as usual. Using this convention, we define the Fourier-Stieltjes transform of a measure μ ∈ M(G).

Definition 103. Let μ ∈ M(G). For each σ ∈� , we define μ̂(σ) to be the operator on H_{σ} defined by

⟨μ̂(σ)ψ, ϕ⟩ = \sqrt{d_{σ}} ∫_{G} \overline{⟨π_{σ}(x)ψ, ϕ⟩}dμ(x).  (27)

The function μ̂ is called the Fourier-Stieltjes transform of μ. For f ∈ L_{1}(G), we define f̂ = μ̂, where dμ = fdx.

From the definition, we have the linearity properties { \atop μ + ν} = μ̂+ν̂ and α̂μ = αμ̂ for all μ, ν ∈ M(G) and α ∈ ℂ. By the orthogonality relations, we see that

⟨π̂_{ij}^{(σ)}(σ′)ψ, ϕ⟩ = \sqrt{d_{σ}} ∫_{G}⟨π_{σ′}(x)ψ, ϕ⟩π_{ij}^{(σ)}(x)dx

= \sqrt{d_{σ}} ∑_{k,l=1}^{d_{σ′}}⟨ψ, ξ_{k}⟩⟨ξ_{l}, ϕ⟩ ∫_{G} π_{kl}^{(σ′)}(x)π_{ij}^{(σ)}(x)dx

= \frac{1}{\sqrt{d_{σ}}} ⟨ψ, ξ_{k}⟩⟨ξ_{l}, ϕ⟩δ_{ki}δ_{lj}δ_{σσ′} = \frac{1}{\sqrt{d_{σ}}} ⟨ψ, ξ_{i}⟩⟨ξ_{j}, ϕ⟩δ_{σσ′}.

By the Peter-Weyl theorem, for any f ∈ L_{2}(G) we have an expansion

f(x) = ∑_{σ∈�} ∑_{i,j=1}^{d_{σ}} d_{σ}^{1/2}f_{ij}^{(σ)}π_{ij}^{(σ)}(x),

where f_{ij}^{(σ)} ∈ ℂ. It follows that

⟨f̂(σ′)ψ, ϕ⟩ = ∑_{σ∈�} ∑_{i,j=1}^{d_{σ}} f_{ij}^{(σ)}⟨ψ, ξ_{i}⟩⟨ξ_{j}, ϕ⟩δ_{σσ′},

so that in particular, the matrix components of f̂(σ) are

⟨f̂(σ)ξ_{i}, ξ_{j}⟩ = f_{ij}^{(σ)}.

Example 104. As an example of finding the Fourier-Stieltjes transform of a function f, we will consider the case when the group G has finite cardinality. In this case, we may enumerate the members of G by G = {g_{i}}_{i=1}^{n}, with g_{1} being the identity for convenience. The group G is a compact topological group under the discrete topology^{12}and every complex-valued function on G is continuous. The (normalized) Haar measure on G is defined by λ(S) = \frac{1}{n}|S| for any arbitrary subset S ⊂ G, where |S| denotes the number of elements contained in S. It follows that any complexvalued function on G is in L_{2}(G), and that L_{2}(G) is isometrically isomorphic with ℂ^{n} as vector spaces under the correspondence f ∈ L_{2}(G) maps to a vector x ∈ ℂ^{n}, whose i’th coordinate x_{i} is given by x_{i} = f(g_{i}) for all 1 ⩽ i ⩽ n. For sake of convenience, we will use f ∈ L_{2}(G) to denote both the function on G and the vector in L_{2}(G), with i’th coordinate f_{i} = f(g_{i}).

Now consider� , and for each σ ∈� fix a representation π_{σ} ∈ σ and a basis {ξ_{i}^{(σ)}}_{i=1}^{d_{σ}} for the representation space of π_{σ}. Additionally, we again let π_{ij}^{(σ)} denote the coordinate map defined by π_{ij}^{(σ)}(g) = ⟨π_{σ}(g)ξ_{i}^{(σ)}, ξ_{j}^{(σ)}⟩. Then for the same reasons as above, π_{ij}^{(σ)} is a vector in ℂ^{n} for all 1 ⩽ i, j ⩽ d_{σ} with k’th coordinate denoted by π_{ijk}^{(σ)} = π_{ij}^{(σ)}(g_{k}). Recall that {π_{ij}^{(σ)} : σ ∈� , 1 ⩽ i, j ⩽ d_{σ}} is an orthonormal basis for L_{2}(G), and therefore for ℂ^{n}. This tells us that the number of elements in the basis must be n, or rather ∑_{σ∈�} d_{σ}^{2} = n. Thus there are finitely many elements in the dual of G, and we enumerate them as� = {σ_{i}}_{i=1}^{r}. Since there will in general be n coordinate maps on G, each of which is representable as a vector in ℂ^{n}, the Fourier-Stieltjes transform will in general be able to be represented as an n × n invertible matrix F, the rows of which are the coordinate maps.

12Let X be a set. The discrete topology for X is the collection of all subsets of X, and with this topology X is a topological space.

As a specific example of this, consider the case when G = S_{3}. We index the elements of G^{13} in the following way:

Element Index
e  1
(12)  2
(13)  3
(23)  4
(123)  5
(132)  6

Them multiplication table for this group is then

g_{1} g_{2} g_{3} g_{4} g_{5} g_{6} g_{1} g_{1} g_{2} g_{3} g_{4} g_{5} g_{6} g_{2} g_{2} g_{1} g_{5} g_{6} g_{4} g_{3} g_{3} g_{3} g_{6} g_{1} g_{5} g_{2} g_{4} g_{4} g_{4} g_{5} g_{6} g_{1} g_{3} g_{2} g_{5} g_{5} g_{3} g_{4} g_{2} g_{6} g_{1} g_{6} g_{6} g_{4} g_{2} g_{3} g_{1} g_{5} .

Let π be a representation of G. We start by assuming that the dimension of the representation space of π is 1, i.e. that π is a homomorphism of G into ℂ. Then necessarily π(g_{1}) = 1, and since g_{5}^{3} = 1, we must have that π(g_{5}) ∈ {ω, ω^{2}, 1}, where ω = exp(2πi/3). Also, g_{2}^{2} = g_{3}^{2} = g_{4}^{4} = e, and therefore π(g_{2}), π(g_{3}), π(g_{4}) ∈ {−1, 1}. Moreover, g_{4}g_{2} = g_{2}g_{3} = g_{3}g_{4} = g_{5}, and it follows immediately that π(g_{2}) = π(g_{3}) = π(g_{4}) and therefore π(g_{5}) = π(g_{2})^{2} = 1. Thus we find exactly two irreducible unitary representations of dimension 1: the mapping 1 : g_{i} ↦ 1 for all 1 ⩽ i ⩽ 6 and the mapping

13We denote the elements of G in the usual cycle notation.

ρ(g_{i}) = \{ \begin{matrix}−1, & i ∈ \{2, 3, 4\}; \\ 1, & otherwise. \end{matrix}

Next, we assume that the dimension of the representation space of π is 2, and see if we can arrive at any new representations that are not reducible into one of the previously found representations. It may be shown that the representation σ defined by

g_{1} ↦ ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) g_{2} ↦ ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} )  g_{3} ↦ ( \begin{matrix}1/2 & −\sqrt{3}/2 \\ −\sqrt{3}/2 & −1/2 \end{matrix} )
g_{4} ↦ ( \begin{matrix}1/2 & \sqrt{3}/2 \\ \sqrt{3}/2 & −1/2 \end{matrix} )g_{5} ↦ ( \begin{matrix}−1/2 & −\sqrt{3}/2 \\ \sqrt{3}/2 & −1/2 \end{matrix} ) g_{6} ↦ ( \begin{matrix}−1/2 & \sqrt{3}/2 \\ −\sqrt{3}/2 & −1/2 \end{matrix} )

This must be all the irreducible unitary representations, since we have two dimension 1 representations and 1 dimension 2 representation and 1^{2} + 1^{2} + 2^{2} = 6 = |G|. We conclude that� = \{1, ρ, \overline{σ}\}, where \overline{σ} represents the equivalence class of the representation σ defined above. The Fourier transform matrix F is therefore given by

F = \frac{1}{|G|}\overline{( \begin{matrix}1(g_{1}) & 1(g_{2}) & 1(g_{3}) & 1(g_{4}) & 1(g_{5}) & 1(g_{6}) \\ ρ(g_{1}) & ρ(g_{2}) & ρ(g_{3}) & ρ(g_{4}) & ρ(g_{5}) & ρ(g_{6}) \\ \sqrt{2}σ(g_{1})_{11} & \sqrt{2}σ(g_{2})_{11} & \sqrt{2}σ(g_{3})_{11} & \sqrt{2}σ(g_{4})_{11} & \sqrt{2}σ(g_{5})_{11} & \sqrt{2}σ(g_{6})_{11} \\ \sqrt{2}σ(g_{1})_{12} & \sqrt{2}σ(g_{2})_{12} & \sqrt{2}σ(g_{3})_{12} & \sqrt{2}σ(g_{4})_{12} & \sqrt{2}σ(g_{5})_{12} & \sqrt{2}σ(g_{6})_{12} \\ \sqrt{2}σ(g_{1})_{21} & \sqrt{2}σ(g_{2})_{21} & \sqrt{2}σ(g_{3})_{21} & \sqrt{2}σ(g_{4})_{21} & \sqrt{2}σ(g_{5})_{21} & \sqrt{2}σ(g_{6})_{21} \\ \sqrt{2}σ(g_{1})_{22} & \sqrt{2}σ(g_{2})_{22} & \sqrt{2}σ(g_{3})_{22} & \sqrt{2}σ(g_{4})_{22} & \sqrt{2}σ(g_{5})_{22} & \sqrt{2}σ(g_{6})_{22} \\ √ & \overline{2} \end{matrix} )}

= \frac{1}{6} ( \begin{matrix}1 & 1 & 1 & 1 & 1 & 1 \\ 1 & −1 & −1 & −1 & 1 & 1 \\ \sqrt{2} & −\sqrt{2} & \sqrt{2}/2 & \sqrt{2}/2 & −\sqrt{2}/2 & −\sqrt{2}/2 \\ 0 & 0 & −\sqrt{6}/2 & \sqrt{6}/2 & −\sqrt{6}/2 & \sqrt{6}/2 \\ 0 & 0 & −\sqrt{6}/2 & \sqrt{6}/2 & \sqrt{6}/2 & −\sqrt{6}/2 \\ \sqrt{2} & \sqrt{2} & −\sqrt{2}/2 & −\sqrt{2}/2 & −\sqrt{2}/2 & −\sqrt{2}/2 \end{matrix} ) .

The matrix components of the Fourier-Stieltjes transform of f in vector form are then given by the matrix product f̂ = F f.

REFERENCES

[1] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis II, Springer-Verlag, New York, NY, 1970.

[2] , Abstract Harmonic Analysis I, Springer-Verlag, New York, NY, 1979.

[3] G. B. Folland, Real Analysis, John Wilen & Sons, New York, NY, 1999.

[4] T. Hungerford, Algebra, Springer, New York, NY, 1974.

[5] U. Krengel, Ergodic Theorems, Walter de Gruyter, Berlin, 1985.

[6] N. Dunford and J. Schwartz, Linear Operators I, John Wilen & Sons, New York, NY, 1966.

[7] R. Howe and E. Tan, Non-abelian Harmonic Analysis, Springer-Verlag, New York, NY, 1992.

INDEX

*-representation, 43

absolutely continuous, 5

measure, 33 adjoint

of a measure, 38

of a linear funtional, 38

character, 66 closed under tensor product, 75 complex measure, 3 conjugate representation, 73 conjugation transformation, 71 continuous

measure, 33 continuous representation, 53 convolution

of functions and measures, 37 of functionals, 27 of measures, 28

coordinate functions, 69 cyclic representation, 44 cyclic vector, 44

discrete measure, 33 dual object, 68 equivalence

of representations, 43

Fourier-Stieltjes transform, 86

Hahn Decomposition, 17

integrable, 5

intertwining operation, 68 invariant vector space, 43 irreducible representation, 43

Jordan Decomposition, 18

Lebesgue-Radon-Nikodym Theorem, 6 left regular representation, 26, 61 linear modulus, 12

maximum and minimum of functions, 1 of linear functionals, 14 multiplicative function, 66 mutually singular

linear functionals, 14 measures, 6

norm of a complex measure, 8 orbit, 44 partition

finite Borel, 3 representation

continuity, 53 cyclic, 44 equivalence, 43 left regular, 26 of a ∗-algebra, 43 of a semigroup, 42 reducibility, 43 unitary, 43

semicharacter, 66 separates points, 76 strongly continuous, 44 tensor product of representations, 74 topological group, 2

locally compact, 2 total variation

of a complex measure, 3 totally bounded, 44 trigonometric polynomials, 70 unitary representation, 43 weakly Borel measureable, 44 weakly continuous, 44