Entropy in type I algebras

Abstract

In the theory of non-commutative entropy the attention has almost exclusively been concentrated on non type I algebras. We shall in the present paper remedy this situation by proving the basic facts on entropy of automorphisms of type I C∗and von Neumann-algebras. The results are as nice as one can hope. The CNT-entropy of an automorphism of a von Neumann algebra of type I with respect to an invariant normal state is the classical entropy of the restriction of the automorphism to the center of the algebra. If one factor of a tensor product of two von Neumann algebras is of type I and the other injective, then the entropy of a tensor product automorphism with respect to an invariant product state is the sum of the entropies. The results have obvious corollaries to type I C∗-algebras. The main idea behind our proofs is the use of conditional expectations of finite index, as employed in [GN]. We shall use the notation hφ(α) for the CNT-entropy of a C ∗-dynamical system as defined by Connes, Narnhofer and Thirring in [CNT], and hφ(α) for the ST-entropy defined by Sauvageot and Thouvenot in [ST].