Download Crossed Products With Continuous Trace by Siegfried Echterhoff PDF

By Siegfried Echterhoff

The significance of separable non-stop hint $C^*$-algebras arises from the next evidence: first of all, their sturdy isomorphism sessions are thoroughly classifiable via topological information and, secondly, continuous-trace $C^*$-algebras shape the development blocks of the extra common kind I $C^*$-algebras. This memoir provides an intensive examine of strongly non-stop activities of abelian in the neighborhood compact teams on $C^*$-algebras with non-stop hint. lower than a few ordinary assumptions at the underlying approach $(A,G,\alpha )$, worthwhile and enough stipulations are given for the crossed product $A{\times }_{\alpha }G$ to have non-stop hint, and a few family among the topological information of $A$ and $A{\times }_{\alpha }G$ are received. the consequences are utilized to enquire the constitution of workforce $C^*$-algebras of a few two-step nilpotent teams and solvable Lie teams.

For readers' comfort, expositions of the Mackey-Green-Rieffel laptop of prompted representations and the idea of Morita identical $C^*$-dynamical platforms are integrated. there's additionally an intensive elaboration of the illustration thought of crossed items by means of activities of abelian teams on style I $C^*$-algebras, leading to a brand new description of activities resulting in kind I crossed items.

Features:

The latest effects at the conception of crossed items with non-stop hint.

Applications to the illustration concept of in the community compact teams and constitution of workforce $C^*$-algebras.

An exposition at the sleek concept of caused representations.

New effects on style I crossed items.

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This allows to translate many results known for ordinary systems to the twisted case. However, the notion of twisted actions used in [47] differs substantially from Green's notion we use in this work. Although it was shown that every twisted action in Green's sense may be transformed into a twisted action in the sense of Packer and Raeburn, this still does not solve all problems, since so far there is no analogue of the modern Mackey-Green machine for the twisted crossed products of [47], which makes it often difficult, if not impossible, to translate results which use the Mackey-Green machine.

E. ) Thus it makes sense to call UJ the Mackey obstruction for extending id to a covariant representation of (/C, G, a, r ) , or just the Mackey obstruction of (/C, G, a, r ) . In any case, if we define Ls = p(c(s)), then L : G —> U is an cj-representation of G such that (1) L n = r n for all n e Nr and (2) a s = A d L s for all s e G. We say that (id, L) is an uj-covariant representation of (/C, G, a, r ) . The following result is the second step of the Mackey machine. Although the result is well known in much more generality (see [33, Theorem 18]) we will present here a very elementary proof, which, in our opinion, gives a clearer picture of the theory than the original proofs.

4). e. M1- = G/M. M(i? T ) which is given by (gM(f)F)(s,i)= f f(r)ar{F(r-\r-H))dr JG and {W™F)(s,i) = mF(s,i) for F e Bl, f € CC(G, A, T) and x € M x . (BT) with the algebra £CT(XT) of all bounded G r -linear operators on T M M X , (g , W ) may be identified with the covariant representation, also denoted (£ M , WM), of (A x a , r G, M \ a) on X T given by QM(m = fH and WxMe = x M ^ ) for all / G C C(G, A , r ) , ^ € l 0 T and x € M x . It was pointed out in [16], following some arguments used in [33, Section 7], that the integrated form gM x WM defines an isomorphism from (A xj a , T G) xi-M- 1 onto BT.

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