Functional Analysis II held by Prof. Dr. Moritz Weber and Dr. Tobias Mai Summer ’18 General information on organisation Tutorials and admission for the final exam To take part in the final exam of this course, 50 % of the total points on all exercise sheets have to be achieved. Furthermore there is compulsory attendence for the tutorials associated to this course, one shouldn’t miss more than two tutorials without good reason. Advise regarding this transcript This transcript is in no way more valuable than a handwritten transcript of the lecture and can by no means replace attendance of the lecture or personal review of the content. Computer composition is no warrant for accuracy! This transcript is written by a student, typos or general mistakes can not be ruled out. Hints towards mistakes in the transcript are explicitly asked for: [email protected] 2 Contents 0 A brief reminder on Hilbert spaces and operators on Hilbert spaces5 1 Locally convex topologies on the space of operators 12 2 Unitisation of C∗-algebras 13 3 Positive elements in C∗-algebras 23 4 Approximate units, ideals, quotients 26 5 Positive linear functionals and the GNS construction 30 6 von Neumann algebras 37 7 The Kaplansky density theorem 42 8 Abelian von Neumann algebras 45 9 Projections, factors and their type classification 51 10 Type I factors and tensor products 57 11 Group von Neumann algebras and type II factors 61 12 The trace construction on finite factors 68 13 The standard representation of tracial von Neumann algebras 72 14 Type III factors 77 15 Universal C∗-algebras 80 16 Example: Toeplitz algebra T 84 17 The (irrational) rotation algebra Aϑ 87 18 Cuntz-Algebra 91 19 Group C∗-algebras 96 20 Products of C∗-algebras 102 3 0 A brief reminder on Hilbert spaces and operators on Hilbert spaces Throughout these lecture notes, the analytic closure in a topological (X, T) for A ⊆ X will be denoted clT(A) or just cl(A), if no confusion is to be feared. Definition 0.1: A pre-Hilbert space (H, h·, ·i) is a K-vector space H (where K = R or K = C) that is endowed with an inner product h·, ·i, i. e. a map h·, ·i : H × H −→ K that satisfies (i) hλx + µy, zi = λhx, zi + µhy, zi for all x, y, z ∈ H, λ, µ ∈ K, (ii) hx, yi = hy, xi for all x, y ∈ H, (iii) hx, xi ≥ 0 for all x ∈ H, (iv) If hx, xi = 0, then x = 0. A Hilbert space (H, h·, ·i) is a pre-Hilbert space (H, h·, ·i) that is complete (i. e. a Banach space) with respect to the norm k·k : H −→ [0, ∞) 1 x 7−→ hx, xi 2 that is induced by the inner product. Remark 0.2 (Properties of pre-Hilbert spaces): Let (H, h·, ·i) be a pre-Hilbert space over K. (i) For all x, y, z ∈ H, λ, µ ∈ K, we have that hz, λx + µyi = λhz, xi + µhz, yi. (ii) k·k is indeed a norm and the Cauchy-Schwarz inequality holds, i. e. for all x, y ∈ H: |hx, yi| ≤ kxkkyk. (iii) k·k satisfies the parallelogram identity, i. e. for all x, y ∈ H: kx + yk2 + kx − yk2 = 2(kxk2 + kyk2) In fact, a normed space is a pre-Hilbert space if and only if the norm satsifies the parallelogram identity. Indeed, the inner product can be recovered by the polarisation identities 3 1 X hx, yi = ikkx + ikyk2 (if K = C), 4 k=0 5 0 A brief reminder on Hilbert spaces and operators on Hilbert spaces 1 hx, yi = (kx + yk2 − kx − yk2) (if K = R). 4 (iv) If x, y ∈ H are orthogonal (x ⊥ y), i. e. hx, yi = 0, then the Pythagorean identiy holds, i. e. kx + yk2 = kxk2 + kyk2. (v) The completion of a pre-Hilbert space (H, h·, ·i) is a Hilbert space. Remark 0.3 (Properties of Hilbert spaces): Let (H, h·, ·i) be a Hilbert space. (i) If C ⊆ H is closed and convex, then for each x ∈ H \ C there is a unique point y0 ∈ C, called the best approximation to x such that kx − y0k = dist(x, C) := inf kx − yk. y∈C (ii) If K ⊆ H is a closed linear subspace, then y0 ∈ K is the best approximation to x ∈ H \ K if and only if hx − y0, yi = 0 for all y ∈ K. (iii) Let M ⊆ H be any subset. We call M ⊥ := {x ∈ H | ∀ y ∈ M : hx, yi = 0} the orthogonal complement of M in H. Note that M ⊥ is a closed linear subspace of H. (iv) Let K1,K2 ⊆ H be closed linear subspaces with K1 ⊥ K2, i. e. hy1, y2i = 0 for all y1 ∈ K1, y2 ∈ K2. Then the (orthogonal) direct sum K1 ⊕ K2 := {y1 + y2 | y1 ∈ K1, y2 ∈ K2} is again a closed linear subspace of H. Furthermore, K1 ∩ K2 = {0} and each y ∈ K1 ⊕ K2 has a unique decomposition y = y1 + y2 with y1 ∈ K1 and y2 ∈ K2. (v) The projection theorem (Theorem 5.18, Functional Analysis I) says that H decomposes as H = K ⊕ K⊥ for any closed linear subspace K ⊆ H; in fact each ⊥ x ∈ H can be written as x = x1 + x2, where x1 ∈ K and x2 ∈ K are the best approximations in K and K⊥ respectively to x. (vi) The Riesz representation theorem (Theorem 5.20, Functional Analysis I) says that the dual space H0 := B(H, K) = {f : H → K linear and bounded} can be identified with H via the anti-linear isometric isomorphism j : H −→ H0 y 7−→ fy := h·, yi. In particular H =∼ H00, i. e. H is reflexive. 6 (vii) A family (ei)i∈I is called orthonormal system, if hei, eji = δi,j for all i, j ∈ I. For such (ei)i∈I , Bessels inequality holds, i. e. for all x ∈ H it holds X 2 2 |hx, eii| ≤ kxk i∈I P with equality if and only if x = i∈I hx, eiiei (Theorem 5.27, Functional Analysis I). We call (ei)i∈I an orthonormal basis, if one of the following equivalent conditions is satisfied:1 (1)( ei)i∈I is a maximal orthonormal system, (2) If x ⊥ ei for all i ∈ I, then x = 0, P (3) For all x ∈ H it holds x = i∈I hx, eiiei, 2 P 2 (4) For all x ∈ H it holds kxk = i∈I |hx, eii| , P K (5) The set { i∈F αiei | F ⊆fin I, αi ∈ } is dense in H. If (fj)j∈J is another orthonormal basis of H, then |I| = |J|; the unique cardinality of an orthonormal basis of H is called the (Hilbert space) dimension of H, denoted by dim H. Every Hilbert space admits an orthonormal basis. Two Hilbert spaces H and K are isomorphic (H =∼ K), i. e. there is a sujective linear map U : H → K that satisfies hUx, UyiK = hx, yiH for all x, y ∈ H, if and only if dim H = dim K. If dim H is countable, we call H separable; this is equivalent to H being separable as a Banach space (i. e. there is a countable dense subset). Example 0.4: (i) Kn is a Hilbert space with the inner product n X > > Kn hx, yi = xjyj for x = (x1, . , xn) , y ∈ (y1, . , yn) ∈ . i=1 (ii) If (Ω, F, µ) is a measure space, then n o L2(Ω, µ) := f :Ω → C measurable : |f(ω)|2 dµ(ω) < ∞ /N , ˆΩ where N := {f :Ω → C measurable | µ({ω ∈ Ω: f(ω) 6= 0}) = 0}, is a Hilbert space with inner product hf, gi = f(ω)g(ω) dµ for all f, g ∈ L2(Ω, µ). ˆΩ 1This is the so called theorem of Parseval, refer to Theorem 5.28 from the Functional Analysis I lecture notes. 7 0 A brief reminder on Hilbert spaces and operators on Hilbert spaces In particular, if a set I is endowed with the counting measure σI , we obtain the 2 2 2 Hilbert space ` (I) := L (I, σI ) with dim ` (I) = |I|. Each separable Hilbert space H that is not finite dimensional satisfies H =∼ `2(N). An orthonormal basis of L2([0, 2π), λ1) with the Lebesgue measure λ1 on [0, 2π) is given by (en)n∈Z where 1 int en(t) = √ e for t ∈ [0, 2π). 2π 2 1 P 2 For f ∈ L ([0, 2π), λ ), the representation f = n∈Zhf, enien (in the L -sense) is called the Fourier series of f and 2π ˆ 1 1 −int 1 fn := √ hf, eni = f(t)e dλ (t) 2π 2π ˆ0 2 P ˆ 2 are called the Fourier coefficients; in particular kfk = 2π n∈Z|fn| . (iii) Hardy spaces and Bergman spaces are Hilbert spaces of holomorphic func- tions, especially so-called reproducing Hilbert spaces. Remark 0.5: Let X be a vector space over K. (i) A map p : X → [0, ∞) is called a seminorm, if it satisfies the following statements: • For all x ∈ X and λ ∈ K it holds p(λx) = |λ|p(x), • For all x, y ∈ X it holds p(x + y) ≤ p(x) + p(y) For x ∈ X and r > 0, we put Bp(x, r) := {y ∈ X | p(y − x) < r}. (ii) We call X • topological vector space, if X is endowed with a topology T with respect to which + : X × X −→ X · : K × X −→ X (x, y) 7−→ x + y (α, x) 7−→ αx are both continuous. • locally convex vector space, if X is a topological vector space whose topology T is generated by some family P of seminorms, i.