Non-commutative Entropy Computations for continuous fields and cross-products

We present here two non-commutative situations where dynamical entropy estimates are possible. The first result is concerned with automorphisms of cross-products by an exact group that commute with the group action and generalizes the result known for amenable group. The second is about continuous fields of C∗-algebras and C(X)automorphisms and shows that one need more than the knowledge of entropy in each fiber. Each result relies on explicit factorization via matrices. Although exact C-algebras have a weaker form of factorization through finite dimensional matrix algebras than nuclear C-algebras, N. Brown [6] was able to show that Voiculescu’s initial definition of non-commutative entropy for automorphisms [15] can be extended to this situation and produced the first computations, notably for automorphisms of cross-products of an exact C-algebra by the group of the integers. Other results, like a striking formula for free product automorphisms of reduced free product C-algebras [7], were obtained proving that it was the correct setting for this kind of entropy. For group C-algebra, Ozawa [12] showed about the same time that the reduced C-algebra of a discrete group is exact if and only if it has an “amenable” action on a compact space. It was long known that this condition was a sufficient one [1]. Indeed since the groupoid associated to the group action on the compact space is amenable, hence its reduced C-algebra is nuclear, the reduced C-algebra of the group is a subalgebra of a nuclear one. Therefore Ozawa result is a geometric caracterization of exactness parallel to Kirchberg’s caracterization of exact C-algebras as subalgebras of nuclear C-algebras. It was then obvious that in the situation of a reduced cross product of an exact C-algebra by an exact group a formula for the entropy of an