By William Arveson
This booklet supplies an advent to C*-algebras and their representations on Hilbert areas. we have now attempted to provide purely what we think are the main simple rules, as easily and concretely as shall we. So every time it truly is handy (and it always is), Hilbert areas turn into separable and C*-algebras turn into GCR. this tradition most likely creates an impact that not anything of price is understood approximately different C*-algebras. after all that's not real. yet insofar as representations are con cerned, we will element to the empirical incontrovertible fact that to at the present time nobody has given a concrete parametric description of even the irreducible representations of any C*-algebra which isn't GCR. certainly, there's metamathematical proof which strongly means that not anyone ever will (see the dialogue on the finish of part three. 4). sometimes, whilst the assumption in the back of the evidence of a basic theorem is uncovered very sincerely in a distinct case, we turn out merely the distinctive case and relegate generalizations to the routines. In impact, we have now systematically eschewed the Bourbaki culture. we've got additionally attempted take into consideration the pursuits of numerous readers. for instance, the multiplicity idea for regular operators is contained in Sections 2. 1 and a couple of. 2. (it will be fascinating yet now not essential to comprise part 1. 1 as well), while somebody attracted to Borel buildings may possibly learn bankruptcy three individually. bankruptcy i may be used as a bare-bones advent to C*-algebras. Sections 2.
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Additional resources for An Invitation to C*-Algebras
The work behind this classification divides naturally into two distinct parts. The first step is to analyze the set of operators in the range of a given representation, and is very general in the sense that the analysis for unitary 40 2. 1 . From Type I to Multiplicity-Free representations of groups is identical with the analysis for representations of C*-algebras, and so on. 1 below. The second step is to classify the multiplicityfree representations of a given object in terms of a suitable set of invariants.
Let x be a self-adjoint element of A. Then the following are equivalent: (i) x O. (ii) There is an element y = y* in A such that x = y2. (iii) There is an element z in A such that x = z*z. PROOF. 1, and (i) implies (ii) is a simple exercise with the functional calculus which we leave for the reader. This result implies an invariance property of the partial order which is often useful: if x ( y then z*xz z*yz for every element z in A. To see why, use (ii) to find u = u* in A with y — x = u2 and note that z*yz — z*xz has the form w*w with w = uz.
CCR and GCR Algebras In the preceding section we saw how one can obtain rather complete information about noncommutative C*-algebras of compact operators. We are now going to introduce a much broader class of C*-algebras. The ideas, and most of the results of this section, originated in a paper of Kaplansky . 5. 1. A CCR algebra is a C*-algebra A such that, for every irreducible representation it of A, it(A) consists of compact operators. 4 implies that every C*-algebra of compact operators is CCR.
An Invitation to C*-Algebras by William Arveson