Entropy of Cuntz’s canonical endomorphism
Entropy of Cuntz’s canonical endomorphism
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DOI：
10.2140/pjm.1999.190.235
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发表时间：
1999-10
影响因子：
0.6
通讯作者：
M. Choda
中科院分区：
文献类型：
--
作者：
M. Choda
cones - st φrmer熵H（·）将Kolmogorov-Sinai的熵不变量推广到有限von Neumann代数的迹保自同构（CS）。用H（·）（[CNT]）的推广定义了C * -代数自同构的cones - narnhoferthirring熵H φ（·），取代了不变态φ的有限迹。这些熵依赖于给定自同构下的不变状态。计算熵的第一个典型有趣的例子是n点集的无限积空间上的伯努利位移βn。在算子代数（von Neumann代数或C * -代数）的情况下，非交换伯努利位移αn代替伯努利位移βn。它是无限张量积A =⊗∞i=−∞Ai（其中Ai是n×n-matrix代数）和H(αn) = logn = hτ (αn) （[CS], [CNT]）上的位移自同构，其中τ是A的唯一迹态，γ是代数B的非周期自同构，那么在交叉积M = B oγ Z中存在一个实现的酉算子u。M的内自同构Adu， （Adu(x) = uxu *）是γ到M的一个扩展。一般来说，γ的熵小于Adu的熵。Stφrmer [S]问γ和Adu的熵是否相等。[j] .核C *代数自同构的拓扑熵ht（·）（cf. [Hu], [T]），它不依赖于任何状态，而是基于近似。作为一个应用，他证明了他的拓扑熵满足伯努利位移βn的等式，因此Connes-Narnhofer-Thirring熵也满足。本文给出了非交换伯努利位移类型的自同构αn和单位*-自同构的等式。
Connes-Stφrmer entropy H(·) extended the entropy invariant of Kolmogorov-Sinai to trace preserving automorphisms of finite von Neumann algebras ([CS]). Replacing a finite trace to an invariant state φ, Connes-NarnhoferThirring entropy hφ(·) is defined for automorphisms of C∗-algebras as a generalization of H(·) ([CNT]). These entropies depend on an invariant state under a given automorphism. The first typical interesting example to compute the entropy is the Bernoulli shift βn on the infinite product space of n-point sets. In the context of operator algebras (von Neumann algebras or C∗-algebras), the non-commutative Bernoulli shift αn takes the place of the the Bernoulli shift βn. It is the shift automorphism on the infinite tensor product A = ⊗∞ i=−∞Ai (where Ai is the n×n-matrix algebra) and H(αn) = logn = hτ (αn) ([CS], [CNT]), where τ is the unique tracial state of A. Ler γ be an aperiodic automorphism of an algebra B. Then there exists an implimenting unitary operator u for γ in the crossed product M = B oγ Z . The inner automorphism Adu, (Adu(x) = uxu∗) of M is an extension of γ to M. In general, the entropy of γ is less than the entropy of Adu. Stφrmer [S] asked if the equality between the entropies of γ and Adu holds. Voiculescu [V] defined topological entropy ht(·) for automorphisms of nuclear C∗-algebras (cf. [Hu], [T]), which does not depend on any state but is based on approximations. As an application, he showed that his topological entropy satisfies the equality for the Bernoulli shift βn, so that Connes-Narnhofer-Thirring entropy does too. In this paper, we show the equality for both of the automorphism αn and the unital *-endomorphism of the type of the non-commutative Bernoulli shift.