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Some Banach Algebra and Operator Theory Notes

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Some Banach Algebra and Operator Theory Notes An Introduction to Operator Algebras Laurent W. Marcoux March 30, 2005 Preface These notes were designed as lecture notes for a first course in Operator Algebras. The student is assumed to have already taken a first course in Linear Analysis. In particular, they are assumed to already know the Hahn- Banach Theorem, the Open Mapping Theorem, etc. A list of those results which will be used in the sequel is included in the second section of the first chapter. March 30, 2005 i Contents Preface i Chapter 1. A Brief Review of Banach Space Theory 1 1. Definitions and examples 1 2. The main theorems 4 Chapter 2. Banach Algebras 7 1. Basic theory 7 2. The functional calculus 19 3. The spectrum 30 Notes for Chapter Two 37 Chapter 3. Operator Algebras 41 1. The algebra of Banach space operators 41 2. The Fredholm Alternative 51 3. The algebra of Hilbert space operators 60 4. The spectral theorem for compact normal operators 66 5. Fredholm theory in Hilbert space 74 Notes for Chapter Three 80 Chapter 4. Abelian Banach Algebras 83 1. The Gelfand Transform 83 2. The radical 89 3. Examples 91 Chapter 5. C*-Algebras 99 1. Definitions and Basic Theory. 99 2. Elements of C∗-algebras. 108 3. Ideals in C∗-algebras. 117 4. Linear Functionals and States on C∗-algebras. 125 5. The GNS Construction. 134 Chapter 6. Von Neumann algebras 139 1. Introduction 139 2. The spectral theorem for normal operators. 145 Appendix A: The essential spectrum 155 155 iii iv CONTENTS Appendix B. von Neumann algebras as dual spaces 161 161 Lab Questions 171 Bibliography 177 Index 179 CHAPTER 1 A Brief Review of Banach Space Theory 1. Definitions and examples 1.1. Definition. A complex normed linear space is a pair (X, k·k) where X is a vector space over C and k · k is a norm on X. That is: • kxk ≥ 0 for all x ∈ X; • kxk = 0 if and only if x = 0; • kλxk = |λ|kxk for all λ ∈ C, x ∈ X • kx + yk ≤ kxk + kyk for all x, y ∈ X. A Banach space is a complete normed linear space. 1.2. Example. Consider (C, k · kp), n ≥ 1, 1 ≤ p ≤ ∞. For x ∈ C, we define n !1/p X p kxkp = |xi| . i=1 We can also consider (C, k · k∞), for each n ≥ 1. In this case, given x ∈ C, we have kxk∞ = max1≤i≤n |xi|. 1.3. Remark. All norms on a finite dimensional Banach space are equivalent. In other words, they generate the same topology. 1.4. Example. Consider (C(X), k · k∞), where X is a compact Haus- dorff space and C(X) = { f : X → C : f is continuous } The norm we consider is the supremum norm, kfk∞ = maxx∈X |f(x)|. 1.5. Example. Let X be a locally compact Hausdorff space. Then (C0(X), k · k∞) is a Banach space, where C0(X) = { f ∈ C(X): f vanishes at ∞ } = { f ∈ C(X): ∀ε > 0, { x ∈ X : |f(x)| ≥ ε } is compact in X }. As before, the norm here is the supremum norm: kfk∞ = supx∈X |f(x)|. 1 2 1. A BRIEF REVIEW OF BANACH SPACE THEORY 1.6. Example. If (X, Ω, µ) is a measure space and 1 ≤ p ≤ ∞, then p L (X, Ω, µ) = { f : X → C : f is Lebesgue measurable Z and |f(x)|p dµ(x) < ∞ } X R p 1/p is a Banach space. The norm here is given by kfkp = X |f(x)| dµ(x) . With (X, Ω, µ) as above, we also define ∞ L (X, Ω, µ) = { f : X → C : f is Lebesgue measurable and for some K ≥ 0, µ({ x ∈ X : f(x) > K }) = 0 }. In this case the norm is kfk∞ = infg=f( a.e. wrt µ) ( supx∈X |g(x)| ) . 1.7. Example. Let I be a set and let 1 ≤ p < ∞. Define `p(I) to be the set of all functions X p { f : I → C : |f(i)| < ∞ } i∈I p P p 1/p p and for f ∈ ` (I), define kfkp = ( i∈I |f(i)| ) . Then ` (I) is a Banach p space. If I = N, we also write ` . Of course, ∞ ` (I) = { f : I → C : sup{ |f(i)| : i ∈ I } < ∞ } and kfk∞ = sup{ |f(i)| : i ∈ I }. A closed subspace of particular interest here is ∞ c0(I) = { f ∈ l (I) : for all ε > 0, card ({ i ∈ I : |f(i)| ≥ ε }) < ∞ }. ∞ Again, if I = N, we write only ` and c0, respectively. 1.8. Example. Consider as a particular case of Example 1.6, T = { z ∈ C : |z| = 1 }, and suppose µ is normalized Lebesgue measure. Then we can define the so-called Hardy spaces Z 2π p p int H (T, µ) = { f ∈ L (T, µ): f(t) exp dt = 0 }. 0 These are Banach spaces for each p ≥ 1, including p = ∞. 1.9. Example. Let n ≥ 1 and let (n) C ([0, 1]) = { f : [0, 1] → C : f has n continuous derivatives }. (k) (n) Define kfk = max0≤k≤n{ sup{ |f (x)| : x ∈ [0, 1] }}. Then C ([0, 1]) is a Banach space. 1.10. Example. If X is a Banach space and Y is a closed subspace of X, then • Y is a Banach space under the inherited norm, and • X/Y is a Banach space — where X/Y = { x + Y : x ∈ X }. The norm is the usual quotient norm, namely: kx+Yk = infy∈Y kx+yk. 1. DEFINITIONS AND EXAMPLES 3 1.11. Example. Examples of Banach spaces can of course be com- bined. For instance, if X and Y are Banach spaces, then we can form the so-called `p-direct sum of X and Y as follows: X⊕pY = { (x, y): x ∈ X, y ∈ Y }, 1/p p p and k(x, y)k = kxkX + kykY . 1.12. Example. Let X and Y be Banach spaces. Then the set of continuous linear transformations B(X, Y) from X into Y is a Banach space under the operator norm k T k = supkxk≤1 kT xk. When X = Y, we also write B(X) for B(X, X). n In particular, B(C ) is isomorphic to the n × n complex matrices Mn and forms a Banach space under a variety of norms, including the operator norm from above. On the other hand, as we observed in Remark 1.3, all such norms must be equivalent to the operator norm. 1.13. Example. Suppose that X is a Banach space. Then X∗ = B(X, C) is a Banach space, called the dual space of X. For example, if µ is σ-finite measure on the measure space (X, Ω), 1 1 1 ≤ p < ∞, and if q, 1 < q ≤ ∞, is chosen so that p + q = 1, then [Lp(X, Ω, µ)]∗ = Lq(X, Ω, µ) [`p]∗ = `q ∗ 1 [c0] = ` . In general, the first two equalities fail if p = ∞. As a second example, suppose X is a compact, Hausdorff space. Then C(X)∗ = M(X) = { µ : µ is a regular Borel measure on X } = { f : X → C : f is of bounded variation on X }. R Note that for Φµ ∈ M(X), the action on C(X) is given by Φµ(f) = X f dµ. For our purposes, one of the most important examples of a Banach space will be: 1.14. Definition. A Hilbert space H is a Banach space whose norm is generated by an inner product h·, ·i, which is a map from H × H → C satisfying: (1) hλx + βy, zi = λhx, zi + βhy, zi (2) hx, yi = hy, xi (3) hx, xi ≥ 0 and hx, xi = 0 if and only if x = 0, 4 1. A BRIEF REVIEW OF BANACH SPACE THEORY 1/2 for all x, y, z ∈ H; λ, β ∈ C. The norm on H is given by kxk = (hx, xi) , x ∈ H. Each Hilbert space H is equipped with a Hilbert space basis. This is an orthonormal set {eα}α∈Λ in H with the property that any x ∈ H P can be expressed as x = α∈Λ xαeα in a unique way. The cardinality of the Hilbert space basis is known as the dimension of the space H. It is a standard result that any two Hilbert spaces of the same dimension are isomorphic. Examples of Hilbert spaces are: n • The space (C , k · k2), n ≥ 1, as defined in Example 1.2. • The space L2(X, Ω, µ) defined in Example 1.6. • The space `2(I) defined in Example 1.7. 2 • The space H (T, µ) defined in Example 1.8. ∗ • Mn is a Hilbert space with the inner product < x, y >= tr(y x). Here ‘tr’ denotes the usual trace functional on Mn, and if y ∈ Mn, then y∗ denotes the conjugate transpose of y. 2. The main theorems 2.1. Theorem. [The Hahn-Banach Theorem] Suppose X is a Ba- nach space, M ⊆ X is a linear manifold and f : M → C is a continuous lin- ear functional. Then there exists a functional F ∈ X∗ such that kF k = kfk and F |M = f. 2.2. Corollary. Let X be a Banach space and suppose 0 6= x ∈ X. Then there exists f ∈ X∗ such that kfk = 1 and f(x) = k x k. 2.3. Corollary. Let X be a Banach space and suppose x 6= y are two vectors in X. Then there exists f ∈ X∗ such that f(x) 6= f(y). 2.4. Corollary. Let X be a Banach space, M be a closed subspace of X and x be a vector in X such that x∈ / M.
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