Entropy for automorphisms of operator algebras

算子代数自同构的熵

• 批准号: 09640185
• 负责人: CHODA Marie
• 金额: $ 1.92万
• 依托单位: Osaka Kyoiku University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640185/
• 关键词: Operator algebras; Automorphism; Non-commutative dynamical system; Entropy; State; CィイD1*ィエD1-algebra; Crossed product; Free product; C^V-環; 制限自由積; 状態(state); 状態(stale)

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.

1.给定一个动力学系统（A,α,φ），我们定义了一个关于φ的熵ht （φ,φ,φ）。对于碳纳米管熵h′′φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ D2（α）和Voiculescu的拓扑熵ht(α)，一般来说h′′φ φ φ φ φ φ D2（α）【小于或等于】ht(α)，但等式并不总是成立。Cuntz的正则内自同态Φof O D2n D2满足h D2[ψ] D2（Φ） = ht d2d2 （Φ），其中ψ是UHF代数的状态。N∧M的Longo正则自同态γ满足h φ φ φ φ φ φ D2(γ) = (1/2)log(IndEγ).2。Alicki-Fannes熵本质上不同于碳纳米管熵。给出了昆兹正则自同态、量子自旋链与位移的交叉积上的内自同态和自由位移的例子。我们从解析和非交换遍历两个角度得到了自由位移的结果。自同构的βB hィイD2φィエD2 (Adu【积】(αβ)= hィイD2φィエD2 (Adu(β),如果B unital,核能,简单,如果交叉产品B×ィイD2βィエD2 Z是简单和纯粹的无限,然后(OィイD2∞ィエD2【积】B)×ィイD2α【积】βィエD2 Z =ィイD4 ~ィエD4 B×ィイD2βィエD2 Z。

项目成果

Nathanial BROWN and Marie CHODA: "Approximation entropies in crossed products with an application to free shifts"Pacific Journal of Mathematics. (to appear).

Nathanial BROWN 和 Marie CHODA：“交叉积中的近似熵及其在自由移位中的应用”《太平洋数学杂志》。

Marie CHODA: "Entropy of Cuntz's canonical endomorphism"Pacific Journal of Mathematics. 190-2. 235-245 (1999)

Marie CHODA：“Cuntz 规范自同态的熵”《太平洋数学杂志》。

Marie CHODA: "Entropy of Cuntz's canonical endomorphism"Pacific Journal of Mathematics. 190. 235-245 (1999)

Marie CHODA：“Cuntz 规范自同态的熵”《太平洋数学杂志》。

J.F.JIANG: "Operator functions associated with the grand Furuta inequality." Math.Inequai.Appl.1・2. 267-277 (1998)

J.F.JIANG：“与大古田不等式相关的算子函数。”Math.Inequai.Appl.1・2（1998）。

Marie Choda: "Endomorphisms of shift type" Operator Algebras and Quantum Field Theory.

Marie Choda：“移位型自同态”算子代数和量子场论。