An Invitation to C*-Algebras by William Arveson

By William Arveson

This publication provides an creation to C*-algebras and their representations on Hilbert areas. we've attempted to offer merely what we think are the main uncomplicated principles, as easily and concretely as shall we. So each time it really is handy (and it always is), Hilbert areas develop into separable and C*-algebras develop into GCR. this tradition most likely creates an impact that not anything of worth is understood approximately different C*-algebras. after all that isn't real. yet insofar as representations are con­ cerned, we will be able to element to the empirical incontrovertible fact that to today 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 facts which strongly means that not anyone ever will (see the dialogue on the finish of part three. 4). sometimes, while the belief in the back of the evidence of a basic theorem is uncovered very truly in a unique case, we turn out in basic terms the distinct case and relegate generalizations to the workouts. In influence, we've systematically eschewed the Bourbaki culture. we have now additionally attempted take into consideration the pursuits of quite a few readers. for instance, the multiplicity concept for regular operators is contained in Sections 2. 1 and a pair of. 2. (it will be fascinating yet now not essential to contain part 1. 1 as well), while an individual attracted to Borel buildings may possibly learn bankruptcy three individually. bankruptcy i'll be used as a bare-bones advent to C*-algebras. Sections 2.

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Let k be the projection of A' onto [sIrie; ]. k belongs to the center of d". We claim first, that SE4 e d, and second, that E; 1 Ec , if C 0 C'. Granting that, it follows that S = E,S;E; is a well-defined element of d (because liSdi tends to 0), and we have C(S) = 4(SE) = (,S'; ) = 7'; (because k, is orthogonal to Yfc , the range of the subrepresentation C, for C' o ). Thus the theorem will be proved. The first claim follows if we show that sIT g_ d for every self-adjoint Te si"; or equivalently, that Td g_ d (since <21 is self-adjoint).

Thus both the ideal W(Ye) and its quotient C*( T)/(e(cle) are CCR algebras of the most tractable kind. Now let A be a general C*-algebra, and let it be an irreducible representation of A on ,lf . Since cocie) is an ideal in "Ye) it follows that the set Wit = {x e A:n(x) E ce (i e° )} is an ideal in A, which contains ker 7E. Of course 23 1. Fundamentals it is possible that W„ = ker it. But in any event the intersection CCR(A) of all these ideals ce„ as it runs over all irreducible representations is an ideal in A, consisting of those elements of A which are compact in every irreducible representation.

Thus U* U and UU* are orthogonal nonzero projections and in particular U and U* do not commute, proving that n(A)/ is not abelian. Conversely, if n(A )' is not abelian then there is a self-adjoint operator T E n(A )' which does not commute with n(A)'; an application of the spectral theorem yields a projection E with the same properties. Thus the range of E cannot be invariant under every operator in n(A )' and so there exists S in Tc(A)' such that (I — E)SE O. Now the initial and final spaces of (I — E)SE are clearly orthogonal, and so the polar decomposition applied to this operator yields a nonzero partial isometry V c ir(A)' such that V* V and V V are orthogonal projections.

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