Entropy for automorphisms of operator algebras

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1.Given CィイD1*ィエD1-dynamical syatem (A,α,φ), we defined an entropy htィイD2φィエD2(α) with respect to φ.For CNT-entropy hィイD2φィエD2(α) and Voiculescu's topological entropy ht(α), in general hィイD2φィエD2(α)【less than or equal】htィイD2φィエD2(α)【less than or equal】ht(α), but the equalities do not always hold. Cuntz's canonical inner endomorphism Φof OィイD2nィエD2 satisfies hィイD2[ψ]ィエD2(Φ) = htィイD2ψィエD2(Φ), where ψ is the state of the UHF algebra. Longo's canonical endomorphisms γfor N ⊂ M satisfies hィイD2ψィエD2(γ) = (1/2)log(IndEγ).2.Alicki-Fannes entropy is essentially different to CNT-entropy. Examples are given as Cuntz's canonical endomorphism, the inner automorphism on the crossed product of a quantum spin chain by the shift, and the free shift.3.We obtained results on the free shift from both analytic and noncommutative ergodic theoretic viewpoints. For an automorphism βof B, hィイD2φィエD2(Adu(α【cross product】β) = hィイD2φィエD2(Adu(β) and if B is unital, nuclear, and simple, and if the crossed product B × ィイD2βィエD2 Z is simple and purely infinite, then (O ィイD2∞ィエD2 【cross product】B) × ィイD2α【cross product】βィエD2 Z =ィイD4〜ィエD4 B × ィイD2βィエD2 Z.