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C-Star Algebras C∗ Algebras Prof. Marc Rieffel notes by Theo Johnson-Freyd UC-Berkeley Mathematics Department Spring Semester 2008 Contents 1: Introduction 4 2: January 23{28, 20084 3: January 30, 20084 3.1 The positive cone......................................5 4: February 1, 20086 4.1 Ideals............................................7 5: February 4, 20089 5.1 Quotient C∗ algebras....................................9 5.2 Beginnings of non-commutative measure theory..................... 10 6: February 6, 2008 11 6.1 Positive Linear Functionals................................ 11 7: February 8, 2008 13 7.1 GNS Construction..................................... 13 8: February 11, 2008 14 9: February 13, 2008 16 9.1 We continue the discussion from last time........................ 17 9.2 Irreducible representations................................. 18 10:Problem Set 1: \Preventive Medicine" Due February 20, 2008 18 1 11:February 15, 2008 19 12:February 20, 2008 20 13:February 22, 2008 22 13.1 Compact operators..................................... 24 14:February 25, 2008 25 15:February 27, 2008 27 15.1 Continuing from last time................................. 27 15.2 Relations between irreducible representations and two-sided ideal........... 28 16:February 29, 2008 29 17:March 3, 2008 31 17.1 Some topology and primitive ideals............................ 31 18:March 5, 2008 32 18.1 Examples.......................................... 32 19:March 7, 2008 35 19.1 Tensor products....................................... 35 20:Problem Set 2: Due March 14, 2008 37 Fields of C∗-algebras....................................... 37 An important extension theorem................................ 38 The non-commutative Stone-Cechˇ compactification...................... 38 Morphisms............................................ 39 21:March 10, 2008 39 21.1 C∗-dynamical systems................................... 39 22:March 12, 2008 41 23:March 14, 2008 41 23.1 Twisted convolution, approximate identities, etc..................... 42 24:March 17, 2008 44 25:March 19, 2008 46 26:March 21, 2008 47 27:March 31, 2008 47 2 28:April 2, 2008 49 28.1 A landscape sketch..................................... 49 28.2 Heisenberg commutation relations............................ 51 29:April 4, 2008 52 30:Problem Set 3: Due April 11, 2008 52 31:April 7, 2008 53 31.1 Some group cohomology.................................. 53 32:April 4, 2008 55 32.1 A specific class of examples................................ 55 33:April 11, 2008 57 34:April 14, 2008 59 35:April 16, 2008 61 35.1 Irreducible representations of algebras Aθ ........................ 61 35.2 Differentiation....................................... 63 36:April 18, 2008 64 36.1 Smooth structures..................................... 64 36.2 Returning to our main example.............................. 66 37:April 21, 2008 67 38:April 23, 2008 69 38.1 We were doing somewhat strange things......................... 69 39:April 25, 2008 72 39.1 Further comments on deformation quantization..................... 72 39.2 Differential forms...................................... 72 39.3 Vector Bundles....................................... 74 40:April 28, 2008 75 40.1 More on vector bundles.................................. 75 41:April 30, 2008 77 42:May 5, 2008 80 42.1 Vector bundles and projective modules.......................... 80 43:May 5, 2008 82 43.1 Some K theory....................................... 82 3 43.2 Return to tori and projective modules.......................... 84 44:May 7, 2008 85 44.1 We go through a careful computation.......................... 85 45:May 9, 2008 87 45.1 Continuation of last time................................. 87 46:May 12, 2008 89 46.1 Guest lecture by W. Arveson: Operator Spaces, \Quantized Functional Analysis".. 90 Index 93 1: Introduction These notes are from the class on C∗-algebras, taught by Prof. Marc Rieffel at UC-Berkeley in the Spring of 2008. The class meets three times a week | Mondays, Wednesdays, and Fridays | from 10am to 11am. I typed these notes mostly for my own benefit | I wanted to try live-TEXing a class, and this one had fewer diagrams and more linear thoughts than my other classes. I do hope that they will be of use to other readers; I apologize in advance for any errors or omissions. Places where I did not understand what was written or think that I in fact have an error will be marked **like this**. Please e-mail me ([email protected]) with corrections. For the foreseeable future, these notes are available at http://math.berkeley.edu/∼theojf/CstarAlgebras.pdf. These notes are typeset using TEXShop Pro on a MacBook running OS 10.5. The raw TEX sources are available at http://math.berkeley.edu/∼theojf/Cstar.tar.gz. These notes were last updated May 12, 2008. 2: January 23{28, 2008 **I had not started taking notes until January 30, 2008. We began with some historical background, then set in on definitions. I may eventually find and type up notes from this first week.** 3: January 30, 2008 Question from the audience: Why are C∗ algebras called that? Answer: They are ∗ algebras, and they are \closed". Von Neuman algebras are \W ∗", because they're \weak-closed". C∗ is also called \spectral". Much less poetic than \sea-star". 4 3.1 The positive cone Last time we stated a big theorem: Theorem: (Fukamya, 1952. Independently in 1953, simpler proof by Kelly and Vaight **spell?**.) ∗ + ∗ + + Let A be a C algebra, A = fa 2 A s.t. a = a and σ(a) 2 R g. Then A is a cone. I.e. if a; b 2 A+ then a + b 2 A+. Key Lemma: For any commutative C(M), M compact, f 2 C(M), f = f¯ the following are equivalent: + (a) f ≥ 0 (i.e. σ(f) 2 R ). (b) For some t ≥ kfk1, kf − t1k ≤ t. (c) For all t ≥ kfk1, kf − t1k ≤ t. Proof of Lemma: (a ) c) Given t ≥ kfk1 and m 2 M, 0 ≥ f(m) − t ≥ t, so jf(m) − tj ≤ t. (c ) b) Obvious. (b ) a) For each m 2 M, jf(m) − tj ≤ t, and t ≥ jf(m)j. Proof of Theorem: Can assure 1 2 A. Given a; b 2 A+, let s = kak, t = kbk. So ka − s1k ≤ s, kb − t1k ≤ t. Then ka + bk ≤ kak + kbk = s + t. Then ka + b − (s + t)1k ≤ ka − s1k + kb − t1k ≤ s + t. So by + Key Lemma, σ(a + b) 2 R . + + Also, A is norm-closed: Say we have an ! a, an 2 A . Choose a t ≥ kank8n. so kan − t1k ≤ t ∗ for all n, and the expression is continuous in the norm, so ka − t1k ≤ t. Moreover, an = an, so a is also self-adjoint, and hence a 2 A+. Theorem: (Kaplansky) If c 2 A, then c∗c ≥ 0. Proof: If not, we can find b = b∗ so that bc∗c = c∗cb and b2c∗c ≤ 0, by taking a bump function. Then set d = cb 6= 0. Then d∗d = b∗c∗cb = bc∗cb = b2c∗c ≤ 0. I.e. we have d 2 A with d∗d ≤ 0; d 6= 0. Now we take real and imaginary parts: d = h + ik, h; k 2 A h∗ = h, k∗ = k. Then d∗d + dd∗ = (h + ik)(h − ik) + (h − ik)(h + ik) = 2(h2 + k2) This is a sum of squares of self-adjoint elements, so is positive. 5 ∗ ∗ ∗ + Ok, but −d d ≥ 0, so dd ≥ 0 is the sum of positive things. So σ(dd ) 2 R , whereas ∗ − σ(d d) 2 R . Prop: Let A be an algebra over field F , and a; b 2 A. Then σ(ab) [ f0g = σ(ba) [ f0g. E.g. Let H be an 1-dim separable Hilbert space, and K and 1-dim subspace, and let S be ∗ ∗ ∗ an isometry of H onto K, then S S = 1H, so σS S = f1g. But SS = orthogonal projection onto K, so σ(SS∗) = f1; 0g. Proof of Prop: a If λ 2 σ(ab), λ 6= 0, i.e. (ab − λ1) is not invertible, so ( λ b − 1) is not invertible. So it suffices to show: if (ab − 1) is invertible, so is (ba − 1). But formally (1 − ab)−1 = 1 + ab + (ab)2 + (ab)3 ::: = 1 + a 1 + ba + (ba)2 + ::: b = 1 + a(1 − ba)−1b and we can check directly that these are inverses. This completes the proof of the theorem. A few final facts: the self-adjoint elements are differences of positive elements, and A+ \ −A+ = f0g. 4: February 1, 2008 Today we present a few simple consequences of this theorem of positivity. Notation: we say a ≤ b is b − a 2 A+ and a and b are self-adjoint. Prop: If 0 ≤ a ≤ b, then kak ≤ kbk. Proof: We can assume 1 2 A. Then b ≤ kbk1, by looking at C∗(b; 1). I.e. kbk1 − b; b − a 2 A+, and + ∗ add, so kbk1 − a 2 A . Then kak ≤ kbk by looking at C (a; 1). Prop: If a ≤ b, then for any c 2 A, we have c∗ac ≤ c∗bc Proof: b − a 2 A+ so b − a = d2 for d ≥ 0. Then c∗bc − c∗ac = c∗(b − a)c = c∗d2c = (dc)∗(dc) ≥ 0. 6 4.1 Ideals We now turn our attention to two-sided ideals of C∗ algebras. These generally do not have identity elements; we will talk about approximate identities. Recall: for a normed algebra A without 1, a left approximate identity for A is a net faλg in A such that eλa ! a for all a 2 A. These don't always exist | stupid counterexample is a Banach space with multiplication always 0. The notions of \right approximate identity" and \two-sided approximate identity" are obvious. We can also ask if our approximate identity is \bounded": does there exist a k so that keλk ≤ k for every λ. Similarly for \approximate identities of norm 1": keλk ≤ 18λ. In the commutative case, Cc(X) of fns with compact support is a dense ideal in C1(X). In 1 the non-commutative case, e.g. B0(H) of compact operators, there are trace-class operators L , Hilbert-Schmidt L2, etc.
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