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 proof: to start with, their sturdy isomorphism sessions are thoroughly classifiable by means of topological info and, secondly, continuous-trace $C^*$-algebras shape the development blocks of the extra normal style I $C^*$-algebras. This memoir offers an intensive research of strongly non-stop activities of abelian in the community compact teams on $C^*$-algebras with non-stop hint. below a few normal assumptions at the underlying procedure $(A,G,\alpha )$, worthy and enough stipulations are given for the crossed product $A{\times }_{\alpha }G$ to have non-stop hint, and a few relatives among the topological facts of $A$ and $A{\times }_{\alpha }G$ are bought. the implications are utilized to enquire the constitution of crew $C^*$-algebras of a few two-step nilpotent teams and solvable Lie teams.

For readers' comfort, expositions of the Mackey-Green-Rieffel computing device of prompted representations and the idea of Morita similar $C^*$-dynamical platforms are incorporated. there's additionally an intensive elaboration of the illustration conception of crossed items by means of activities of abelian teams on variety I $C^*$-algebras, leading to a brand new description of activities resulting in sort I crossed items.

Features:

The newest effects at the thought of crossed items with non-stop hint.

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

An exposition at the smooth concept of triggered representations.

New effects on variety I crossed items.

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MAXIMALLY POINTWISE UNITARY SUBGROUPS PROOF. 9 that ind£(pxW|Ep) = Affp/Sp®(pxW), where denotes the left regular representation of Hp/Yp. \HP/XP —" '— Since A# / S FT is weakly equivalent to Hp/Yp we conclude that ind s p ( p x W | s , ) is weakly equivalent to Hp/Yp r ifp)~. But this implies that ker(ind# (p x xW)) = ker(ind# (p x W)) for all x £ Hp/Yp. Thus it follows from the continuity of induction and the theorem of induction in stages that k e r ( i n d g > x W\*p)) = ker(indg p (ind£;(p x W\xp))) = Q ^ k e r ( i n d g > x XW)) = ker(indg p (p x W)).

This is especially true if we want to use results known for ordinary crossed products, but for which no twisted version is available in the literature. This problem was partly solved by Packer and Raeburn in [47] by showing that every twisted action is exterior equivalent to an ordinary action. 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.

Thus ^(L 1 (G , / , C, rf,)QJC) is dense in L X (G, /C, r) since <£' has dense image in Ll(G,JC). Now let i£' be a unitary representation of G" such that R' restricted to T is a multiple of the identity character and let R denote the u~l-representation of G defined by R$ = R',s Xy Then we compute (1HR ® id) x (R 0 £,)(*(/ 0 k)) = I 1HR ® i d ( * ( / 0 k)(s))R5 0 L 5 ds = / / ( s , l)(fl* ®k)ds= (f / ( 5 , l)fl( 5|1) ds J 0 fc = # 0 id(/ 0 fc) for all / 0 k e Ll(G", C, r") 0 /C. This completes the proof.

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