Von Neumann Algebras for Abstract Harmonic Analysis by Cameron Zwarich A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Master of Mathematics in Pure Mathematics Waterloo, Ontario, Canada, 2008 © Cameron Zwarich 2008 I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Abstract This thesis develops the theory of operator algebras from the perspective of abstract harmonic analysis, and in particular, the theory of von Neumann algebras. Results from operator algebras are applied to the study of spaces of coefficient functions of unitary representations of locally compact groups, and in particular, the Fourier algebra of a locally compact group. The final result, which requires most of the material developed in earlier sections, is that the group von Neumann algebra of a locally compact group is in standard form. iii Acknowledgements First of all, I would like to thank my supervisors, Brian Forrest and Nico Spronk. Without them, this thesis would not have been possible. I would like to thank my readers, Hun Hee Lee and Peter Wood, who spent a considerable amount of time reading a draft of this lengthy thesis. Finally, I would like to thank my fellow graduate students who participated in the harmonic analysis learning seminars, where much of the material in this thesis was originally presented. iv Contents 1 Introduction 1 2 Banach ∗-algebras 3 2.1 Definitions and Basic Properties . 3 2.2 Commutative C∗-algebras . 11 2.3 Positivity and Order in C∗-algebras . 15 2.4 Bounded Approximate Identities . 20 2.5 Homomorphisms, Ideals, and Quotients . 27 2.6 Unitaries, Projections, and Partial Isometries . 29 2.7 Operator Topologies . 36 2.8 Density Theorems . 56 2.9 Representations . 73 2.10 Positive Linear Functionals . 86 2.11 Pure Positive Functionals and Irreducible Representations . 109 2.12 Positive Linear Functionals on C∗-algebras . 113 3 Von Neumann Algebras 123 3.1 Basic Properties . 123 3.2 Comparison of Projections . 127 3.3 Normal Linear Functionals . 131 3.4 Normal maps . 138 3.5 The Enveloping von Neumann Algebra of a C∗-algebra . 143 3.6 The Polar Decomposition of Normal Functionals . 155 3.7 The Radon-Nikodym Theorem for Normal Functionals . 159 4 The Fourier Algebra 167 4.1 Definition and Basic Properties . 167 4.2 Herz’s Restriction Theorem . 174 4.3 Standard Form and the Fourier Algebra . 178 References 191 v Chapter 1 Introduction This thesis is intended to be an introduction to abstract harmonic analysis from the modern viewpoint of operator algebras. This begins with a treatment of the abstract theory of C∗-algebras and their connection to the representa- tion theory of locally compact groups. It continues with the detailed develop- ment of revelant portions of the theory of von Neumann algebras. It concludes with an introduction to Fourier algebra of a general locally compact group, in- cluding Eymard’s theorem on the spectrum, Herz’s restriction theorem, and the fact that a group von Neumann algebra is always in standard form. When I first began the study of abstract harmonic analysis, I soon realized that the literature is very scattered. In order to gain sufficiency in the subject to the point where one can read the recent research literature, one must learn from a number of different sources, spanning distinct historical periods and notational conventions. There is no one source that one can read for an intro- duction to the subject, and some important results can be difficult to locate in their most useful form. I set out to alleviate this problem and provide one source from which one can learn the fundamentals of the subject. This thesis presumes a good working knowledge of functional analysis, particularly the duality theory of Banach spaces, a basic knowledge of Banach algebras, and a passing knowledge of integration theory on locally compact groups. However, besides these prerequisites, every argument is developed in full, often with more details than the proofs in the original papers or in other sources. While this thesis is quite comprehensive in that it gives full proofs for all of the subjects that it covers, there are many relevant topics that are not covered. In particular, there are no serious examples of group representations, and there is no mention of weak containment of representations. The main reason for their exclusion is the lack of space, as this thesis is already well over the average length of a Master’s thesis in mathematics. The type decomposition of von Neumann algebras was also excluded on the same grounds. 1 I consulted many sources in the preparation of this thesis. Besides the pa- pers cited explicitly, I read most of the standard textbooks on the subject of operator algebras: Dixmier’s two books [Dix69a] [Dix69b], Takesaki’s introduc- tory textbook [Tak02], Pedersen’s book [Ped79], the book of Stratila and Zsido [SZ79], and Davidson’s book [Dav96]. While all of the results in this thesis ap- pear in one cited source or another, the proofs given here are often a synthesis of the different ideas that I have learned while studying the subject as a whole, rather than being taken directly from one particular place. Since my goal was to give a synthetic treatment of the subject, looking back into the past rather than developing the subject genetically, I do not discuss the attribution of each theorem as it is proved. This is not meant to take credit away from those who toiled to prove it, rather it is meant to provide a more coherent narrative. Of course, there are exceptions made when a theorem is generally referred to by the name of the person who proved it. At the end of each section, I have written some notes on the history of the topic covered in that section, and these notes often provide additional information about the particular proofs that I chose to present. 2 Chapter 2 Banach ∗-algebras 2.1 Definitions and Basic Properties 2.1.1 Definition. Let A be an algebra over C. An involution on A is a map ∗ : A!A such that for all a; b 2 A and λ 2 C, (i) (a∗)∗ = a, (ii) (a + b)∗ = a∗ + b∗, (iii) (λa)∗ = λa∗, (iv) (ab)∗ = b∗a∗. If A is equipped with an involution, it is called a ∗-algebra. By a unital ∗- algebra we mean a ∗-algebra with a multiplicative unit 1 such that 1 ≠ 0. 2.1.2 Examples. (i) The complex numbers form a ∗-algebra, where the involution is conjuga- tion. (ii) Let X be a locally compact Hausdorff space. Then the algebras Cc(X), C0(X),Cb(X), and C(X) of continuous functions on X are ∗-algebras, where the involution is pointwise conjugation. (iii) Let H be a Hilbert space. Then the algebra B(H ) of bounded linear oper- ators on H is a ∗-algebra, where the involution is the adjoint operation. (iv) Let G be a locally compact group, and ∆ its modular function. Then 1 the algebras Cc(G),L (G), and M(G), all equipped with the convolution product, are ∗-algebras, where the measure µ∗ is defined by Z Z g(s) dµ∗(s) = g(s−1) dµ(s) G 1 for f 2 C0(G). If f 2 L (G) and µ is the measure defined by Z Z g(s) dµ(s) = f (s)g(s) ds; 3 then Z Z Z g(s) dµ∗(s) = f (s)g(s−1) ds = ∆(s−1)f (s−1)g(s). Hence f ∗(s) = ∆(s−1)f (s−1). 2.1.3 Definition. Let A be a ∗-algebra. If a 2 A is such that a = a∗, we say that a is self-adjoint. We let Asa denote the set of self-adjoint elements of A. We say that a 2 A is normal if aa∗ = a∗a, unitary if a is invertible and a−1 = a∗, and a projection if a is self-adjoint and a2 = a. 2.1.4 Examples. (i) The self-adjoint elements of the C are the real numbers. Every element of C is normal. The unitary elements of C are the elements of norm 1. The projections in C are 0 and 1. (ii) Let X be a locally compact Hausdorff space. If f 2 C0(X), then f is always normal (as C0(X) is commutative); f is unitary if and only if ran(f ) ⊆ fλ 2 C : jλj = 1g, and f is a projection if and only if ran(f ) ⊆ f0; 1g, in which case f is the characteristic function of a clopen set. (iii) Let H be a Hilbert space. The self-adjoint elements of B(H ) are the self-adjoint or Hermitian operators, i.e. those bounded operators T such that hT ξ j ηi = hξ j T ηi for all ξ; η 2 H . If T 2 B(H ), then T is normal, unitary, or a projection if and only if it is normal, unitary, or an orthog- onal projection in the usual meaning of those words for operators. In particular, U 2 B(H ) is unitary if and only if hUξ j Uηi = hξ j ηi for all ξ; η 2 H and U has dense range. It is important to note that if A is a unital ∗-algebra, then the unit 1 is a self-adjoint element of A. This is because 1∗ = 1∗1 = (1∗1)∗ = 1∗∗ = 1: It is clear that Asa is a real vector subspace of A.