By Siegfried Echterhoff

ISBN-10: 0821805630

ISBN-13: 9780821805633

The significance of separable non-stop hint $C^*$-algebras arises from the next proof: to begin with, their solid isomorphism periods are thoroughly classifiable by way of topological facts and, secondly, continuous-trace $C^*$-algebras shape the development blocks of the extra common style I $C^*$-algebras. This memoir offers an in depth research of strongly non-stop activities of abelian in the community compact teams on $C^*$-algebras with non-stop hint. below a few typical assumptions at the underlying process $(A,G,\alpha )$, beneficial 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 got. the consequences 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 precipitated representations and the speculation of Morita similar $C^*$-dynamical structures are integrated. there's additionally an in depth elaboration of the illustration thought of crossed items via activities of abelian teams on kind I $C^*$-algebras, leading to a brand new description of activities resulting in style I crossed items.

Features:

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

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

An exposition at the glossy idea of caused representations.

New effects on kind I crossed items.

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**Extra resources for Crossed Products With Continuous Trace**

**Example text**

Suppose now that ir x U G (A xa G)~such that P — ker(7r x U). Then ir x U is a GCi^-element of (A xa G^by assumption. Thus {TT X U} is a Ginvariant locally closed subset of Prim(A xa G). Let B denote the corresponding subquotient of A xa G. Then B x^G is a locally closed subset of (A xa G) x^G. 2). To this end suppose that p' is an element of A such that 46 3. REPRESENTATIONS OF T Y P E I ABELIAN T W I S T E D SYSTEMS INDp' G (B x^Gy. 2 that ker(7r xU) = ker (resf lc} (INDp')) = ker(indf e} p'), from which follows that p' is in the quasi-orbit of p.

The group G*-algebra C*(N) is type I) closed normal subgroup of the second countable locally compact group G such that G/N is abelian. Then C*(G) is isomorphic to the twisted crossed product C*(N) X 7 N J T N G of the abelian twisted system (C*(N), G, 7 N , rN). If V G N, then a closed subgroup HofGis maximally V-unitary with respect to the twisted action ( 7 ^ , rN) if and only if H is maximal with respect to the property that there exists a unitary extension, say V, of V to H. 3. Let G be a separable locally compact group and let N be a closed normal type I subgroup of G such that G/N is abelian.

7. Then the following conditions are equivalent: (1) A x a ? T G is type I (resp. CCR). (2) The G-orbit G(p x U) is locally closed (resp. closed) in (A xa^T Hpy for all pxU ell. (3) All Mackey obstructions of (A, G, a, r ) are type I and the G-orbits G(a x W) are locally closed (resp. closed) in (A x a > r Spy for all p G A and a x W G (A x a , T Spy such that ker<7 = kerp. Moreover, if A x a ? T G is type I then ind# (p x U) is irreducible for all pxU e1Z and ind# (p x U) is equivalent to ind# , (p' x U') for some other p' x U' G 11, if and only if p' x Uf = (p x U) o 7S p for some s G G.

### Crossed Products With Continuous Trace by Siegfried Echterhoff

by Paul

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