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MTH 510/615: Operator Theory and Operator Algebras Semester 2, 2018-19

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MTH 510/615: Operator Theory and Operator Algebras Semester 2, 2018-19 Dr. Prahlad Vaidyanathan Contents 1 Banach Algebras3 1.1 Definition and Examples...........................3 1.2 Invertible Elements..............................7 1.3 Spectrum of an Element........................... 10 1.4 Unital Commutative Banach Algebras................... 15 1.5 Examples of the Gelfand Spectrum..................... 19 1.6 Spectral Permanence Theorem........................ 21 1.7 Exercises.................................... 23 2 C*-Algebras 27 2.1 Operators on Hilbert Spaces......................... 27 2.2 Spectrum of an Element........................... 36 2.3 Unital Commutative C∗ algebras...................... 39 2.4 Spectrum of a Normal Operator....................... 44 2.5 Positive Operators and Polar Decomposition................ 49 2.6 Exercises.................................... 52 3 The Spectral Theorem 53 3.1 The Finite Dimensional Case........................ 53 3.2 Multiplication Operators........................... 56 3.3 The Spectral Theorem............................ 61 3.4 Exercises.................................... 65 3.5 Complex Measures.............................. 66 3.6 Borel Functional Calculus.......................... 71 3.7 Spectral Measures............................... 82 3.8 Compact Normal Operators......................... 84 3.9 Exercises.................................... 89 4 Additional Topics 91 4.1 Quotients of C∗ algebras........................... 91 4.2 Positive Linear Functionals.......................... 94 4.3 The Gelfand-Naimark-Segal Construction.................. 101 5 Instructor Notes 107 2 1 Banach Algebras 1.1 Definition and Examples All vector spaces in this course will be over C. Definition 1.1.1. 1. An algebra is a vector space A over C together with a bilinear multiplication under which A is a ring. In other words, for all α; β 2 C; a; b; c 2 A, we have (αa + βb)c = α(ac) + β(bc) and a(αb + βc) = α(ab) + β(ac) 2. An algebra A is said to be a normed algebra if there is a norm on A such that a)( A; k · k) is a normed linear space b) For all a; b 2 A, we have kabk ≤ kakkbk 3. A Banach algebra is a complete normed algebra. Remark 1.1.2. 1. If X is a normed linear space, then kx + yk ≤ kxk + kyk. Hence, the map (x; y) 7! x + y is jointly continuous. ie. If xn ! x and yn ! y, then xn + yn ! x + y. 2. Similarly, if A is a normed algebra, then the map (x; y) 7! xy is jointly continuous [Check!] Example 1.1.3. 1. A = C 2. A = C[0; 1]. More generally, C(X) for X A compact, Hausdorff space. A = Cb(X), where X is a locally compact Hausdorff space. 3. A = C0(X), where X is a locally compact Hausdorff space. [Exercise] 4. A = c0, the space of complex sequences converging to 0. Note: All the above examples are abelian. 3 5. A = Mn(C) for any n 2 N with the operator norm n kAk = supfkA(x)k : x 2 C ; kxk ≤ 1g 6. A = B(X) for any Banach space X. 7. A = L1(R) with multiplication given by convolution Proof. For f; g 2 A, we write Z f ∗ g(x) := f(t)g(x − t)dt R Now Z Z Z kf ∗ gkL1(R) = jf ∗ g(x)jdx ≤ jf(t)jjg(x − t)jdtdx = kfkkgk R R R by Fubini's theorem. The other axioms are easy to check. 8. A = `1(Z) with multiplication given by convolution (proof is identical to the previous one). A is a commutative Banach algebra. Definition 1.1.4. Let A be a Banach algebra. 1. A subset I ⊂ A is called a left ideal of A if it is a vector subspace of A and a 2 A; b 2 I ) ab 2 I 2. A right ideal is defined similarly. 3. In this course, an ideal will refer to a two-sided ideal, for which we write I C A. 4. An ideal I C A is said to be proper if I 6= f0g and I 6= A. 5. A maximal ideal is an ideal that is not properly contained in any proper ideal. Example 1.1.5. 1. A = C[0; 1], then I = ff 2 C(X): f(1) = 0g is a maximal ideal. 2. If A = Mn(C), then A has no non-trivial ideals Proof. Let f0g 6= J C A, then choose 0 6= T 2 J, then 9Ti;j = a 6= 0. Let Ek;l be the permutation matrix obtained by switching the kth row of the identity matrix with the lth row. Then 0 T := E1;jTEi;1 2 J 0 th and T1;1 = a 6= 0. Now let F1;1 be the matrix with 1 in the (1; 1) entry and zero elsewhere. Then 1 F T 0F = F 2 J a 1;1 1;1 1;1 Similarly, F2;2;F3;3;:::;Fn;n 2 J. Adding them up, we have ICn 2 J and since J is an ideal, this means that J = A. 4 (End of Day 1) 3. If X is a locally compact Hausdorff space, then C0(X) is an ideal in Cb(X), the space of bounded continuous functions on X 4. Let X be a Banach space and A = B(X), then set F(X) = fT 2 B(X): T has finite rankg Then F(X) is an ideal in A. 5. If A = B(X), then the set K(X) of compact operators on X is a closed ideal in A. In fact, if H is a Hilbert space, then K(H) = F(H) Theorem 1.1.6. If A is a Banach algebra, and I C A is a proper closed ideal, then A=I is a Banach algebra with the quotient norm ka + Ik = inffka + bk : b 2 Ig Proof. 1. Clearly, A=I is an algebra. 2. Now we check that the axioms of the norm hold : a) If ka + Ik = 0, then 9bn 2 I such that ka + bnk ! 0. Since I is closed, this means that a 2 I and hence a + I = 0 in A=I b) Clearly, ka + Ik ≥ 0. c) If a; b 2 A, then for any c; d 2 I ka + b + Ik ≤ ka + b + c + dk ≤ ka + ck + kb + dk This is true for any c; d 2 I, so taking infimum gives ka + b + Ik ≤ ka + Ik + kb + Ik 3. We now check that A=I is a Banach algebra: If a; b 2 A, then for any c; d 2 I we have (a + c)(b + d) = ab + cb + ad + dc where x := cb + ad + dc 2 I. Hence kab + Ik ≤ kab + xk ≤ ka + ckkb + dk This is true for any c; d 2 I, so taking infimum gives kab + Ik ≤ ka + Ikkb + Ik 4. A is complete (See [Conway, Theorem III.4.2]) Definition 1.1.7. Let A and B be Banach algebras. 1. A map ' : A ! B is called a homomorphism of Banach algebras if 5 a) ' : A ! B is a continuous (ie. bounded) linear transformation of normed linear spaces b) '(ab) = '(a)'(b) for all a; b 2 A. 2. Recall that if ' is continuous, then k'k = supfk'(a)k : a 2 A; kak ≤ 1g 3. A bijective homomorphism whose inverse is also continuous is called an isomorphism of Banach algebras. Example 1.1.8. 1. If I C A is a closed ideal, then the natural quotient map π : A ! A=I is a homomorphism. Note that kπ(a)k = ka + Ik ≤ ka + 0k = kak Hence kπk ≤ 1. We will see later (Theorem 1.2.9) that kπk = 1 if A is unital. 2. If A = C(X) and x0 2 X, then f 7! f(x0) is a continuous homomorphism. 3. Let A be any Banach algebra, and B(A) be the space of bounded linear operators on A. Define ' : A !B(A) by x 7! Lx; where Lx(y) := xy Then ' is a continuous homomorphism, called the left regular representation of A. 4. Let X = [0; 1], H = L2(X) and set A = C(X);B = B(H). Define ' : A ! B by f 7! Mf where Mf (g) := fg. Then ' is a continuous homomorphism. (End of Day 2) Theorem 1.1.9. Let ' : A ! B be a homomorphism of Banach algebras and let I = ker('). Then 1. I = ker(') is a closed ideal in A 2. There is a unique injective homomorphism ' : A=I ! B such that ' ◦ π = '. Furthermore, k'k = k'k 6 Proof. We know from algebra that 9 a unique homomorphism of rings ' : A=I ! B such that ' ◦ π = ' which is given by '(a + I) = '(a) It is easy to see that ' is linear as well, and so is a homomorphism of algebras. Further- more, for any c 2 I k'(a + I)k = k'(a)k = k'(a + c)k ≤ k'kka + ck Taking infimum, we see that ' is continuous and k'k ≤ k'k. However, since kπk ≤ 1 by Example 1.1.8(1), k'k = k' ◦ πk ≤ k'kkπk ≤ k'k Hence, k'k = k'k. 1.2 Invertible Elements Definition 1.2.1. Let A be a Banach algebra 1. A is said to be unital if 9e 2 A such that ae = ea = a for all a 2 A. 2. If A is unital with unit e, then we will write 1A = 1 = e, and assume that k1Ak = 1. 3. If A is unital, then we may assume C ⊂ A via the map α 7! α1A Remark 1.2.2. ([Arveson, Theorem 1.4.2]) Let (A; k·k) be a complex algebra with a unit e that is also a Banach space. Furthermore, assume that the multiplication map (x; y) 7! xy is jointly continuous.
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