Entropy of Automorphisms of AF Algebras

Abstract

Here we consider a notion of topological entropy for automorphisms of AF algebras, based on a corresponding measure theoretic entropy of Connes and St0rmer [6] for automorphisms of hyperfinite von Neumann algebras with an invariant trace. We show how to compute the entropy of the shift on certain AF algebras associated with topological Markov chains. If jtf is a unital AF algebra, let T(jtf) denote the normalized traces on $/. If 4> E T(jtf), and B is a finite dimensional C*-subalgebra of $#, let E% denote the conditional expectation of stf onto B, relative to 4>. For simplicity, we will always only consider finite dimensional subalgebras, with the same unit as stf. If n e N9 let Sn denote the maps x from Z into j^+ with finite support such that £ x(al5..., an) = l. For l<Z<n, xESn, aeZ, put afeZ