Topological entropy and crossed products
by
Ciprian Pop
IMAR and Texas A&M University
Coauthors: Roger A. Smith (Texas A&M University)

The topological entropy was introduced by Adler, Konheim and McAndrew. It is a positive constant associated in a suitable way to any continuous morphism of a compact Hausdorff space. Its noncommutative counterpart was defined by Voiculescu, for the case of automorphisms of nuclear C^*-algebras. If X is a compact Haussdorff space and T is a homeomorphism on X, then the classical topological entropy h(T) agrees with ht(\alpha_T), Voiculescu's topological entropy of the induced automorphism of C(X).

Later on, N. Brown generalized this notion for automorphisms of exact C^*-algebras. He also proved that, given (A, \alpha, G) a C^*-dynamical system where A is exact and G is a discrete commutative group, then for any g in G,

htA(\alphag)=htA×\alpha G(Ad\lambdag),

where Ad_\lambdag denotes the inner automorphism of A×_\alpha G induced by \alpha_g.

By using a completely new technique we generalize Brown's result to the case when G is an amenable, locally [FIA]^- group. The latter class of groups is a superclass of [FIA]^-, in particular contains the class of commutative as well as compact groups.

Date received: May 27, 2002