Dorte Olesen

Pointwise unitary automorphism groups

Abstract Let ( A , G , α ) be a separable C ∗ -dynamical system in which G is abelian and A has continuous trace. The action α is said to pointwise unitary if for each π in A there is a covariant representation ( π , U ) of the system ( A , G , α ). It is shown that for pointwise unitary actions the spectrum ( A × α G ) ∧ of the crossed product is a locally compact Hausdorff space and the restriction map Res, which takes π × U in ( A × G ) ∧ to π, maps ( A × G ) ∧ onto A . Furthermore, Res induces a homeomorphism of ( (A × α G) ∧ G onto A . These results lead to a characterization of crossed products A × α G with continuous trace: when A has continuous trace and G acts freely on A , this happens precisely when the action of G on A is also proper. This extends Greens characterization of free and proper transformation groups. Finally, it is shown that the results have applications to the problem of determining the dual topology on a locally compact group.

Compact abelian groups of automorphisms of simpleC*-algebras

If α is a continuous representation of a compact abelian groupG as *-automorphisms of a prime or simpleC*-algebraA we study how algebraic properties of the fixed-point algebraA0 are reflected in properties of the spectrum of α and outerness of α1,t∈G. For example, ifA is prime, Γ(α)=Sp(α) if and only ifA0 is prime, and the same with simple (instead of prime) whenG is finite. In the latter case the relative commutant ofA0 inA is trivial if and only if all α1≠l are outer, and ifG is cyclic of prime order all the conditions above are equivalent. This paper replaces the earlier versions [12] and [9].

Groups of automorphisms with spectrum condition and the lifting problem

AbstractWe show that a continuous one-parameter group {αt} of automorphisms of a separableC*-algebraA satisfies a spectrum condition if and only if it is the limit, pointwise onA, uniformly on compact subsets of IR of a sequence of inner automorphism groups whose positive generators do not increase too fast. Moreover we prove in this case that if π is a surjective morphism of a separableC*-algebraB onA then there is a similar group

Ergodic actions of compact abelian groups

\{ \bar \alpha _t \}

C^*

of automorphisms ofB such that

The Connes spectrum for actions of Abelian groups on continuous-trace algebras

\pi \circ \bar \alpha _t = \alpha _t \circ \pi