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 subsequent evidence: to start with, their good isomorphism periods are thoroughly classifiable through topological facts 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 community compact teams on $C^*$-algebras with non-stop hint. lower than a few common assumptions at the underlying procedure $(A,G,\alpha )$, worthy and adequate stipulations are given for the crossed product $A{\times }_{\alpha }G$ to have non-stop hint, and a few family among the topological info of $A$ and $A{\times }_{\alpha }G$ are got. the implications 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 computing device of brought on representations and the idea of Morita identical $C^*$-dynamical platforms are integrated. there's additionally an intensive elaboration of the illustration conception of crossed items by way of activities of abelian teams on variety I $C^*$-algebras, leading to a brand new description of activities resulting in style I crossed items.

Features:

The most up-to-date effects at the concept of crossed items with non-stop hint.

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

An exposition at the sleek concept of precipitated representations.

New effects on sort I crossed items.

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The general case for non-trivial r and a follows now by passing over to the quotient X x UjT L of X xu L. 2. Morita equivalence and duality for abelian twisted s y s t e m s If (A, G, a, T) is a twisted covariant system such that G = G/NT is abelian, then we call (A, G, a, r ) an abelian twisted system, or just an abelian system if r is trivial. 2. MORITA EQUIVALENCE AND DUALITY 35 for all x € G and / G CC(G, A , T ) . a is called the dual action of G on A xi a>r G. Suppose now that M is a closed subgroup of G containing NT.

The quotient space QG(M) consisting of 24 1. PRELIMINARIES AND BASIC DEFINITIONS all G-quasi orbits in M is called the quasi-orbit space of M. 221 and Corollary 19]. 13. Suppose that (A, G, a, r ) is a separable twisted covariant system. Then the map P —> Q(P) sending P G Prim(A) to its quasi-orbit is open and the map QG(Piim(A)) - 1(A); Q(P) -> ker Q(P) = f] sec as(P) is a homeomorphism onto its image. Moreover, for any ix x U G (A x a ? r GJ" there exists a unique quasi-orbit ResG(7r x U) G Q(3(Prim(A)) such that ker-zr = kerResG(7r x U) and the resulting map Res G : (A x a > r Gy —• Q<3(Prim(>l)) is continuous.

Let Y be the symmetry group of (/C, G, a, r ) . Then Indg : (/C x a , T E)T-> Prim(X: x Q , r G);p x V -> ker(indg(p x V)) is a homeomorphism with inverse given by the restriction map Res^ : Prim(/C x ft)T G) —> Prim(/C XQ, E) = (/C x a E)", which is defined by Res s(ker(7r x U)) = ker(7r x E/|E). Moreover, for each P 6 Prim(/C x a > r G), tiie map G —> Prim(/C x a ) T G ) ; x ^ ^ x ^ ) Actors through a homeomorphism between Y and Prim(/C x a ? r G). In particular, each P G Prim(/C x a ? r G) has constant stabilizer Y1- under the action of G.

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