Entropy in type I algebras

TL;DR

This paper demonstrates that for type I von Neumann algebras, the entropy of a dynamical system equals the entropy on its center, and for injective systems, the entropy of tensor products sums up.

Contribution

It establishes a precise relationship between entropy in type I algebras and their centers, and extends to tensor products of injective systems.

Findings

Entropy equals the entropy on the center for type I algebras.

Tensor product system entropy equals the sum of individual entropies for injective systems.

Provides a clear formula for entropy in specific W*-dynamical systems.

Abstract

It is shown that if (M,phi,alpha) is a W*-dynamical system with M a type I von Neumann algebra then the entropy of alpha w.r.t. phi equals the entropy of the restriction of alpha to the center of M. If furthermore (N,psi,beta) is a W*-dynamical system with N injective then the entropy of the tensor product system is the sum of the entropies.