# Crossed Products With Continuous Trace by Siegfried Echterhoff

By Siegfried Echterhoff

The significance of separable non-stop hint $C^*$-algebras arises from the next evidence: first of all, their strong isomorphism periods are thoroughly classifiable via topological information and, secondly, continuous-trace $C^*$-algebras shape the construction blocks of the extra normal style I $C^*$-algebras. This memoir offers an in depth examine of strongly non-stop activities of abelian in the community compact teams on $C^*$-algebras with non-stop hint. less than a few common assumptions at the underlying procedure $(A,G,\alpha )$, precious and adequate stipulations are given for the crossed product $A{\times }_{\alpha }G$ to have non-stop hint, and a few kinfolk among the topological information of $A$ and $A{\times }_{\alpha }G$ are got. the consequences are utilized to enquire the constitution of staff $C^*$-algebras of a few two-step nilpotent teams and solvable Lie teams.

For readers' comfort, expositions of the Mackey-Green-Rieffel desktop of triggered representations and the speculation of Morita an identical $C^*$-dynamical platforms are integrated. there's additionally an in depth elaboration of the illustration thought of crossed items via activities of abelian teams on style I $C^*$-algebras, leading to a brand new description of activities resulting in variety I crossed items.

Features:

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

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

An exposition at the sleek idea of precipitated representations.

New effects on kind I crossed items.

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Z^^a. 4. 10. i&R extends to a *-homomorphism from G 0 (O,A) onto A which is G-equivariant with respect to the diagonal action of G on Co (ft, A). ) = potyR is equal to the pair (R(p),p) G O x A = LEMMA c0(n,Ar. Suppose now that (A, G, a, r) is a twisted covariant system and that H is a closed subgroup of G such that there exists a continuous G-equivariant map R from Prim(j4) onto G/H. Then it is easily seen that *&R transports f onto r . Thus, \$R implements a canonical surjective *-homomorphism, say \£, from BT = C0(G/H, A) x ^ f G onto A x a >T G.

CCR) if and only if the G-orbit G(W) is locally closed (resp. closed) in Hy for all W G 71. If this is true, then ind^ v W is irreducible for all W G 71, and induction defines a bijection between the G-orbits in 7Z and G. A locally compact group G is called monomial if every irreducible representation of G is induced from a one-dimensional representation of some subgroup. Recall also that a group G is called metabelian (or two-step solvable) if the commutator subgroup [G, G] of G is abelian. 4.

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. 13, Prim(f? x ^ G) consists of only one G-orbit with respect to the double dual action a, we conclude that the orbit of ker p' in Prim(A) is locally closed.