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Notes on Operator Algebras

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Notes on Operator Algebras Notes on operator algebras Jesse Peterson April 6, 2015 2 Contents 1 Spectral theory 7 1.1 Banach and C∗-algebras . 8 1.1.1 Examples . 13 1.2 The Gelfand transform . 14 1.3 Continuous functional calculus . 16 1.3.1 The non-unital case . 17 1.4 Applications of functional calculus . 19 1.4.1 The positive cone . 20 1.4.2 Extreme points . 22 2 Representations and states 25 2.1 Approximate identities . 25 2.2 The Cohen-Hewitt factorization theorem . 26 2.3 States . 29 2.3.1 The Gelfand-Naimark-Segal construction . 31 2.3.2 Pure states . 33 2.3.3 Jordan Decomposition . 35 3 Bounded linear operators 37 3.1 Trace class operators . 40 3.2 Hilbert-Schmidt operators . 44 3.3 Compact operators . 47 3.4 Topologies on the space of operators . 50 3.5 The double commutant theorem . 53 3.6 Kaplansky's density theorem . 56 3.7 The spectral theorem and Borel functional calculus . 57 3.7.1 Spectral measures . 58 3.8 Abelian von Neumann algebras . 63 3 4 CONTENTS 3.9 Standard probability spaces . 67 3.10 Normal linear functionals . 75 3.11 Polar and Jordan decomposition . 77 4 Unbounded operators 83 4.1 Definitions and examples . 83 4.1.1 The spectrum of a linear operator . 86 4.1.2 Quadratic forms . 87 4.2 Symmetric operators and extensions . 89 4.2.1 The Cayley transform . 91 4.3 Functional calculus for normal operators . 94 4.3.1 Positive operators . 94 4.3.2 Borel functional calculus . 95 4.3.3 Polar decomposition . 99 4.4 Semigroups and infinitesimal generators. 100 4.4.1 Contraction semigroups . 100 4.4.2 Stone's Theorem . 102 4.4.3 Dirichlet forms . 103 I Topics in abstract harmonic analysis 105 5 Basic concepts in abstract harmonic analysis 107 5.1 Polish groups . 107 5.2 Locally compact groups . 110 5.3 The L1, C∗, and von Neuman algebras of a locally compact group . 114 5.3.1 Unitary representations . 117 5.4 Functions of positive type . 121 5.5 The Fourier-Stieltjes, and Fourier algebras . 127 5.6 Pontryagin duality . 132 5.6.1 Subgroups and quotients . 141 5.6.2 Restricted products . 142 5.6.3 Stone's theorem . 142 5.7 The Peter-Weyl Theorem . 144 5.8 The Stone-von Neumann theorem . 147 CONTENTS 5 6 Group approximation properties 153 6.1 Ergodicity and weak mixing . 153 6.1.1 Mixing representations . 154 6.2 Almost invariant vectors . 156 6.3 Amenability . 158 6.4 Lattices . 164 6.4.1 An example: SLn(Z) < SLn(R) . 164 6.5 The Howe-Moore property for SLn(R) . 167 6.6 Property (T) . 169 6 CONTENTS Chapter 1 Spectral theory If A is a complex unital algebra then we denote by G(A) the set of elements which have a two sided inverse. If x 2 A, the spectrum of x is σA(x) = fλ 2 C j x − λ 62 G(A)g: The complement of the spectrum is called the resolvent and denoted ρA(x). Proposition 1.0.1. Let A be a unital algebra over C, and consider x; y 2 A. Then σA(xy) [ f0g = σA(yx) [ f0g. Proof. If 1 − xy 2 G(A) then we have (1 − yx)(1 + y(1 − xy)−1x) = 1 − yx + y(1 − xy)−1x − yxy(1 − xy)−1x = 1 − yx + y(1 − xy)(1 − xy)−1x = 1: Similarly, we have (1 + y(1 − xy)−1x)(1 − yx) = 1; and hence 1 − yx 2 G(A). Knowing the formula for the inverse beforehand of course made the proof of the previous proposition quite a bit easier. But this formula is quite natural to consider. Indeed, if we just consider formal power series then we have 1 1 X X (1 − yx)−1 = (yx)k = 1 + y( (xy)k)x = 1 + y(1 − xy)−1x: k=0 k=0 7 8 CHAPTER 1. SPECTRAL THEORY 1.1 Banach and C∗-algebras A Banach algebra is a Banach space A, which is also an algebra such that kxyk ≤ kxkkyk: An involution ∗ on a Banach algebra is a conjugate linear period two anti- isomorphism such that kx∗k = kxk, for all x 2 A. An involutive Banach algebra is a Banach algebra, together with a fixed involution. If an involutive Banach algebra A additionally satisfies kx∗xk = kxk2; for all x 2 A, then we say that A is a C∗-algebra. If a Banach or C∗-algebra is unital, then we further require k1k = 1. Note that if A is a unital involutive Banach algebra, and x 2 G(A) then −1 ∗ ∗ −1 ∗ (x ) = (x ) , and hence σA(x ) = σA(x). Lemma 1.1.1. Let A be a unital Banach algebra and suppose x 2 A such that k1 − xk < 1, then x 2 G(A). P1 k Proof. Since k1 − xk < 1, the element y = k=0(1 − x) is well defined, and it is easy to see that xy = yx = 1. Proposition 1.1.2. Let A be a unital Banach algebra, then G(A) is open, and the map x 7! x−1 is a continuous map on G(A). Proof. If y 2 G(A) and kx − yk < ky−1k then k1 − xy−1k < 1 and hence by the previous lemma xy−1 2 G(A) (hence also x = xy−1y 2 G(A)) and 1 X kxy−1k ≤ k(1 − xy−1)kn n=0 1 X 1 ≤ ky−1knky − xkn = : 1 − kyk−1ky − xk n=0 Hence, kx−1 − y−1k = kx−1(y − x)y−1k ky−1k2 ≤ ky−1(xy−1)−1kky−1kky − xk ≤ ky − xk: 1 − ky−1kky − xk ky−1k2 Thus continuity follows from continuity of the map t 7! 1−ky−1kt t, at t = 0. 1.1. BANACH AND C∗-ALGEBRAS 9 Proposition 1.1.3. Let A be a unital Banach algebra, and suppose x 2 A, then σA(x) is a non-empty compact set. x Proof. If kxk < jλj then λ − 1 2 G(A) by Lemma 1.1.1, also σA(x) is closed by Proposition 1.1.2, thus σA(x) is compact. ∗ To see that σA(x) is non-empty note that for any linear functional ' 2 A , −1 we have that f(z) = '((x−z) ) is analytic on ρA(x). Indeed, if z; z0 2 ρA(x) then we have −1 −1 −1 −1 (x − z) − (x − z0) = (x − z) (z − z0)(x − z0) : Since inversion is continuous it then follows that f(z) − f(z0) −2 lim = '((x − z0) ): z!z0 z − z0 We also have limz!1 f(z) = 0, and hence if σA(x) were empty then f would be a bounded entire function and we would then have f = 0. Since ' 2 A∗ were arbitrary this would then contradict the Hahn-Banach theorem. Theorem 1.1.4 (Gelfand-Mazur). Suppose A is a unital Banach algebra ∼ such that every non-zero element is invertible, then A = C. Proof. Fix x 2 A, and take λ 2 σ(x). Since x − λ is not invertible we have that x − λ = 0, and the result then follows. Pn k If f(z) = k=0 akz is a polynomial, and x 2 A, a unital Banach algebra, Pn k then we define f(x) = k=0 akx 2 A. Proposition 1.1.5 (The spectral mapping formula for polynomials). Let A be a unital Banach algebra, x 2 A and f a polynomial. then σA(f(x)) = f(σA(x)). Pn k Proof. If λ 2 σA(x), and f(z) = k=0 akz then n X k k f(x) − f(λ) = ak(x − λ ) k=1 k−1 X X j k−j−1 = (x − λ) ak x λ ; k=1 j=0 10 CHAPTER 1. SPECTRAL THEORY hence f(λ) 2 σA(x). conversely if µ 62 f(σA(x)) and we factor f − µ as f − µ = αn(x − λ1) ··· (x − λn); then since f(λ) − µ 6= 0, for all λ 2 σA(x) it follows that λi 62 σA(x), for 1 ≤ i ≤ n, hence f(x) − µ 2 G(A). If A is a unital Banach algebra and x 2 A, the spectral radius of x is r(x) = sup jλj: λ2σA(x) Note that by Proposition 1.1.3 the spectral radius is finite, and the supre- mum is attained. Also note that by Proposition 1.0.1 we have the very useful equality r(xy) = r(yx) for all x and y in a unital Banach algebra A. A priori the spectral radius depends on the Banach algebra in which x lives, but we will show now that this is not the case. Proposition 1.1.6 (The spectral radius formula). Let A be a unital Banach n 1=n algebra, and suppose x 2 A. Then limn!1 kx k exists and we have r(x) = lim kxnk1=n: n!1 Proof. By Proposition 1.1.5 we have r(xn) = r(x)n, and hence r(x)n ≤ kxnk; n 1=n showing that r(x) ≤ lim infn!1 kx k . n 1=n To show that r(x) ≥ lim supn!1 kx k , consider the domain Ω = fz 2 C j jzj > r(x)g, and fix a linear functional ' 2 A∗. We showed in Propo- sition 1.1.3 that z 7! '((x − z)−1) is analytic in Ω and as such we have a Laurent expansion 1 X an '((z − x)−1) = ; zn n=0 for jzj > r(x).
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