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 elemental leads to operator algebras. effects mentioned comprise Gelfand's illustration of commutative C*-algebras, the GNS development, 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. routines are supplied after every one bankruptcy.

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7 DEFINITION a in C*-Algebra Suppose A is a (a) We say that x is self-adjoint (b) We that x is unitary say (c) We say that (d) We say that (e) We say that C*-algebra and x is = x. = xx* = 1, or equivalently in A. if x* if x* x x is normal if x*x = xx*. ) in = X-I. x* A. are normal. It is also elements It is clear that both self-adjoint and unitary and are positive. We clear that positive elements are self-adjoint projections element is a unit vector. write x > 0 if and only if x is positive. 1 THEOREM If x is normal assume that x is First PROOF self-adjoint.

Let Co(O) be the of complex-valued functions on 0 that can be uniformly approximated by continuous functions on 0 with compact The space Co(0) is sometimes support. space to as the space of continuous functions on 0 which vanish at 00, because a continuous function on 0 if to and if for > 0 Co f belongs only every \342\202\254 (0) there exists a compactset O\342\202\254 in 0 such that for all x E 0 - O\342\202\254. < \342\202\254 If (x) If I o == 0 U {(X)} is the one-point of then a function in 0, compactification f C(O) belongs to Co(O) if and only if f( (0) == O.

And hence is is multiplicative cpz Define I(n)zn L n=-oo) k=-oo Thus k)zn-k n=-oo) 00) = I(n - L k=-oo = - k)g(k) I(n L k=-oo) n=-CX) MLI(Z) maximal the in = (z) by show We cpz. ideal space of \302\2431 (Z). that is a surjective homeomorphism. For eachn in Z let In be the function in \302\2431 defined by In(n) = 1 and (Z) = 0 for k n. It is easy to check the following f n (k) =1= properties: elementary 10 = 1, Il/nll = 1, and In * Ik = In+k for all nand k in Z. That z from follows is one-to-one To see that On the other let is onto, 1 = hand, E ML, cp II < Izol.

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