# An Introduction to Operator Algebras by Kehe Zhu

By Kehe Zhu

An creation to Operator Algebras is a concise text/reference that specializes in the basic ends up in operator algebras. effects mentioned comprise Gelfand's illustration of commutative C*-algebras, the GNS building, the spectral theorem, polar decomposition, von Neumann's double commutant theorem, Kaplansky's density theorem, the (continuous, Borel, and L8) practical calculus for regular operators, and kind decomposition for von Neumann algebras. workouts are supplied after every one bankruptcy.

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Example text

Yx) = 1. 5 COROLLARY Suppose A is a Banachalgebra a(xy) and U {O} x, YEA. 6 SupposeA is a Banach alRebra and x, YEA. 1 Let T be the = on H defined operator Tf(x) \302\2432 [0, xE = xf(x), 1] by [0,1]. is the spectrum of T in B (H)? Generalize your result. Hilbert space. Show that H is a (separable)infinite dimensional Suppose for every nonempty compact set K in the complex plane there exists an = K. 5 If x the spectrum that a(T*) is invertible of T = in {X in defined a Banach 1 x be an {O, ai, a2, .

In N if x yx)2, x(yxy) + (yxy)x) is. Since) (xy- 'IIx)2 + (XU + lIx)2 = 2 [x(yxy) + (yxy)x]))) that the 26) Linear Multiplicative Functionals) and) - (cp(xy yx))2 = cp((xy is in N for all x we see that xy - yx and xy + yx, we concludethat xy belongs particular, N is closedunder multiplication, yx)2), and yEA. Adding to N for all x in N and y the proof of the completing E N xy - in A. yx In theorem. 1 Show M n (C) has no nontrivial ideals. that maximal ideals. 3 Convince yourself linear functionals on in D.

It is easy to see that the closure of I is again of I we conclude that I is equal to its closure and in A. By the maximality hence I is closedin A. Since A is commutative and I is maximal, the quotient is a division (a general fact from abstract algebra algebra). 2),there exists 4> : A/I \037 C which is an isometric isomorphism. Let 7r be the quotient mapping from A onto A/I, then 4> 0 7r is a multiplicative on A with kernel I. linear functional to that the above linear see between correspondence multiplicative Finally, functionals and maximal ideals is one-to-one,let