A C*-Dynamical Entropy and Applications to Canonical Endomorphisms

✍ Scribed by Marie Choda

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 217 KB
• Volume: 173
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.2000.3564

✦ Synopsis

For an automorphism : of a unital C*-algebra A, we give a definition of an entropy ht , (:) with respect to an :-invariant state , of A. For Connes Narnhofer Thirring entropy h , (:) and Voiculescu's topological entropy ht(:), in general h , (:) ht , (:) ht(:), but the equalities do not always hold.

We compute entropies of an endomorphism \ with respect to the state . defined from a left inverse of . Cuntz's canonical inner endomorphism 8 of O n satisfies h . (8)=ht . (8), which is determined by the mean entropy of . on the UHF (uniformly hyperfinite) algebra. If # is Longo's canonical endomorphism for an irreducible graded standard AFD (approximately finite dimensional) inclusion N/M of infinite factors with finite index, then h . (#)=(1Â2) log(Ind E # ), for the conditional expectation E # on #(M).