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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Additional resources for Crossed Products With Continuous Trace

Example text

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.