By William Arveson
This booklet supplies an advent to C*-algebras and their representations on Hilbert areas. now we have attempted to offer purely what we think are the main uncomplicated principles, as easily and concretely as lets. So each time it truly is handy (and it always is), Hilbert areas turn into separable and C*-algebras develop into GCR. this custom most likely creates an impact that not anything of worth is understood approximately different C*-algebras. in fact that's not real. yet insofar as representations are con cerned, we will element to the empirical proven fact that to at the present time not anyone 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 at the back of the facts of a basic theorem is uncovered very basically in a different case, we turn out in basic terms the designated case and relegate generalizations to the routines. In impression, we have now systematically eschewed the Bourbaki culture. we've got additionally attempted take into consideration the pursuits of quite a few readers. for instance, the multiplicity idea for regular operators is contained in Sections 2. 1 and a pair of. 2. (it will be fascinating yet no longer essential to contain part 1. 1 as well), while anyone attracted to Borel buildings may possibly learn bankruptcy three individually. bankruptcy i'll be used as a bare-bones creation to C*-algebras. Sections 2.
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Additional info for An Invitation to C*-Algebras
5 shows that each operator T of the type discussed above is a GCR operator in the following sense. 6. A Hilbert space operator T is called a GCR operator if C*(T) is a GCR algebra. Since finitely generated C*-algebras are always separable composition series associated with a GCR operator is countable. The class of all GCR operators contains the normal operators, the compact operators, and a great variety of others as well. Moreover, we will see in the sequel that it is the GCR operators (and they alone) which lend themselves to the possibility of analysis in terms of "generalized" spectral theory and multiplicity theory.
Isomorphic with A/ker it and therefore it contains a nonzero CCR ideal. 4) this implies in particular that it(A) contains a nonzero compact operator. 2. Now let it and a be two irreducible representations of A on_ spaces ,Ye° and At', respectively, such that ker it = ker a. 1: it(x) i — u(x), x e A, defines an irreducible representation of rc(A). 4 is irreducible. 11w(ye) is equivalent to the identity representation of W(le). 4. again, it follows that A itself is equivalent to the identity representation of ir(A).
1. is unitarily equivalent to the identity representation, 0 and note that the latter implies it — a. PROOF. Note in particular that an irreducible representation of a CCR algebra is determined (to equivalence) by its kernel. 4). There are many common C*-algebras which are not CCR. Here is a simple example. Let T be an irreducible operator on a Hilbert space If which is not compact but whose imaginary part is compact. Then C*(T) is not CCR, for the identity representation of C* (T) is irreducible but its range contains the noncompact operators T and I.
An Invitation to C*-Algebras by William Arveson