Entropy for canonical shifts

Choda, Marie

doi:10.1090/s0002-9947-1992-1070349-0

Cited by 19 publications

(18 citation statements)

References 12 publications

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“…From the lemma above and theorem 3.4, we get the following conclusion, which is also proved in [1] and [2] by different methods without appealing to theorem 3.4.…”

Section: And Thatsupporting

confidence: 70%

Endomorphisms and automorphisms from subfactors illustrating non-commutative entropy

Svendsen¹

2005

MATH. SCAND.

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We present a series of examples of endomorphisms and automorphisms arising from subfactors, which illustrate some of the recent theorems in non-commutative entropy theory. Moreover it is shown that for these examples the Connes-Størmer entropy of the automorphism is maximal and coincides with the topological entropy. This follows from the theorems our examples illustrate, and is also showed directly.

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“…From the lemma above and theorem 3.4, we get the following conclusion, which is also proved in [1] and [2] by different methods without appealing to theorem 3.4.…”

Section: And Thatsupporting

confidence: 70%

Endomorphisms and automorphisms from subfactors illustrating non-commutative entropy

Svendsen¹

2005

MATH. SCAND.

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“…Then it is known that lim n→∞ 2 n H(M ∩ M n ) exists and is equal to the Connes-Størmer dynamical entropy of the canonical shift on (R, φ). (See [3,5] for definition and properties of the canonical shift. )…”

Section: Preliminariesmentioning

confidence: 99%

“…This notion has turned out to play a fundamental role when we study group actions on N ⊂ M. Roughly speaking, an automorphism on N ⊂ M is strongly outer if and only if it does not appear in the descendant sectors (or bimodules) of N ⊂ M. (See Proposition 1.11 below.) Also, it should be mentioned that there are many close connections between subfactor theory and entropy theory; for instance, the relation between the index [M : N ] and the relative entropy H(M |N ) [36] (also [13]), the dynamical entropy of the canonical shift [3,5], the characterization of the strong amenability of G N,M in terms of the relative entropy [16,39], etc.…”

Section: Introductionmentioning

confidence: 99%

Standard invariants for crossed products inclusions of factors

Hiai¹

1997

Pacific J. Math.

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When N ⊂ M is an inclusion of factors with finite index and a group G acts on N ⊂ M, we compare the standard invariants of N ⊂ M and the crossed product inclusion N G ⊂ M G. The cases when G is a discrete group and when G is a locally compact abelian group are separately considered. Applying to a common crossed product decomposition, we obtain comparison results between the type II and type III standard invariants of an inclusion of type III factors. Introduction.Since Jones [21] initiated the index theory of type II 1 subfactors, a great progress has been made particularly on classification of hyperfinite type II 1 subfactors (see [35,[37][38][39] and also [20,23] In the course of classifying hyperfinite type II 1 subfactors with small indices, it has been observed that symmetries on principal graphs (or paragroup symmetries) play a vital role typically in orbifold constructions (see [4,8,10,20,23]). A paragroup symmetry is an action of a (finite) group G on the principal graph and Ocneanu's connection made from an inclusion N ⊂ M. When this symmetry can extend to the subfactor level, the crossed product inclusion N G ⊂ M G arises and its standard invariant 237

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“…Various entropies of the Jones shifts have been already calculated, and we will recall some of the results here. (In connection with the index theory for type II l subfactors, one can find deeper results on the entropies in [3], [4] and [6], viewing the Jones shifts as the square roots of the canonical shifts. )…”

Section: (V Mmentioning

confidence: 99%

A Note on the Approximation Entropies of Certain Shifts

Nishiuchi

1997

Publ. Res. Inst. Math. Sci.

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Voiculescu has proposed several routes to quantum entropies. Among them, the notion of the "approximation entropies" is a group of four entropies with similar definitions, based on two kinds of approximations. The C*-cases are extensions of the classical topological entropy and the W*-cases are those of the measure-theoretic one. In this paper, we will focus on the approximation entropies and investigate the entropies of Powers' binary shifts with some condition and the Jones shifts. §1. Preliminaries Voiculescu [15] has introduced quantum entropies of automorphisms of operator algebras, called approximation entropies. In this section, we will review the definitions and fix the notations. Using two kinds of approximation for the W*-case and the C*-case, Voiculescu has defined four approximation entropies which have similar definitions to each other. As Voiculescu said, one may think of approximation entropies as "growth"-entropies and the key concept is the "(5-rank". The four approximation entropies are defined in the same way, as a matter of form, except for the "(5-rank". So we will only state the definition by subalgebra approximation for the W*-case in detail, (see [15] for the other cases.)Let Ji be a hyperfinite von Neuman algebra with a faithful normal tracial state i and 3F(JC) be the set of the unital finite-dimensional C*-subalgebras of M. By &f(Ji) we denote the set of finite subsets of Ji and by Aut(^) the automorphism group of Jt. For a normal faithful state cp on M, set AutpT, (p) = {a e AutpT) | cp ° a = cp}. For CD e &f(Jf) and X a Ji, we shall write Communicated by Y.

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Entropy for canonical shifts

Cited by 19 publications

References 12 publications

Endomorphisms and automorphisms from subfactors illustrating non-commutative entropy

Endomorphisms and automorphisms from subfactors illustrating non-commutative entropy

Standard invariants for crossed products inclusions of factors

A Note on the Approximation Entropies of Certain Shifts

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