It is shown that if (M,phi,alpha) is a W*-dynamical system with M a type I von Neumann algebra then the entropy of alpha w.r.t. phi equals the entropy of the restriction of alpha to the center of M. If furthermore (N,psi,beta) is a W*-dynamical system with N injective then the entropy of the tensor product system is the sum of the entropies.
Noncommutative topological entropy of endomorphisms of Cuntz algebras II
Cuntz代数的规范不变自同态的非交换拓扑熵的研究开始于我们早期的工作与Joachim Zacharias是继续和扩展到自同态不一定是置换型。特别地，证明了如果H是N维希尔伯特空间，V是H与自身的张量积上的不可约乘法酉，F是张量翻转，则与酉VF相关联的Longo正则自同态的Voiculescu熵等于log N.
All generating sets of all property T von Neumann algebras have free entropy dimension $\leq 1$
设$N$是一个扩散性质的T von Neumann代数，X是N的自伴元的任意有限生成集.通过使用刚性/变形参数适用于表示N在全矩阵代数，我们推导出X的微观状态空间是渐近离散的酉共轭。我们用这个描述来证明X的自由熵维数$δ_0(X)$小于或等于1。因此，当N嵌入到超有限$\mathrm{II}_1$ -因子的超积中时，则为$δ_0(X)=1$，否则为$δ_0(X)=-\infinity$。这推广了Voiculescu和Ge，Shen关于$SL_n(\mathbb Z)$的早期结果以及Connes，Shlyakhtenko关于任意性质T代数的群生成元的结果。
We show that the (Gurevich) topological entropy for the countable Markov shift associated with an infinite transition matrix $A$ coincides with the non-commutative topological entropy for the Exel--Laca algebra associated with $A$, under certain conditions on $A$. An important example satisfying the conditions is the renewal shift, which is not locally finite. We also pose interesting questions for future research on non-commutative topological entropy for non-locally finite transition matrices.
For $p\geq 2$, the $p$-adic ring $C^*$-algebra $\mathcal{Q}_p$ is the universal $C^*$-algebra generated by a unitary $U$ and an isometry $S_p$ such that $S_pU=U^pS_p$ and $\sum_{l=0}^{p-1}U^lS_pS_p^*U^{-l}=1$. For any $k$ coprime with $p$ we define an endomorphism $χ_k\in{\rm End}(\mathcal{Q}_p)$ by setting $χ_k(U):=U^k$ and $χ_k(S_p):=S_p$. We then compute the entropy of $χ_k$, which turns out to be $\log |k|$. Finally, for selected values of $k$ we also compute the Watatani index of $χ_k$ showing that the entropy is the natural logarithm of the index.
We consider Pimsner algebras that arise from C*-correspondences of finite rank, as dynamical systems with their rotational action. We revisit the Laca-Neshveyev classification of their equilibrium states at positive inverse temperature along with the parametrizations of the finite and the infinite parts simplices by tracial states on the diagonal. The finite rank entails an entropy theory that shapes the KMS-structure. We prove that the infimum of the tracial entropies dictates the critical inverse temperature, below which there are no equilibrium states for all Pimsner algebras. We view the l…
We study the equilibrium simplex of Nica-Pimsner algebras arising from product systems of finite rank on the free abelian semigroup. First we show that every equilibrium state has a convex decomposition into parts parametrized by ideals on the unit hypercube. Secondly we associate every gauge-invariant part to a sub-simplex of tracial states of the diagonal algebra. We show how this parametrization lifts to the full equilibrium simplices of non-infinite type. The finite rank entails an entropy theory for identifying the two critical inverse temperatures: (a) the least upper bound for existence…
Conditioning on Subalgebras - Entropy Duality and Generalized Quantum Stein's Lemma in von Neumann Algebras
Department of Applied Mathematics and Theoretical Physics, University of Cambridge（剑桥大学应用数学与理论物理系）；School of Mathematics and Statistics, Wuhan University（武汉大学数学与统计学院）；Department of Mathematical Sciences, Chalmers University of Technology（查尔姆斯理工大学数学科学系）
In the framework of Quantum Field Theory, we provide a rigorous, operator algebraic notion of entanglement entropy associated with a pair of open double cones $O \subset \tilde O$ of the spacetime, where the closure of $O$ is contained in $\tilde O$. Given a QFT net $A$ of local von Neumann algebras $A(O)$, we consider the von Neumann entropy $S_A(O, \tilde O)$ of the restriction of the vacuum state to the canonical intermediate type $I$ factor for the inclusion of von Neumann algebras $A(O)\subset A(\tilde O)$ (split property). We show that this canonical entanglement entropy $S_A(O, \tilde O…
We study quantum computation relations on unital finite-dimensional CAR $C^{*}$-algebras. We prove an entropy power inequality (EPI) in a fermionic setting, which presumably will permit understanding the capacities in fermionic linear optics. Similar relations to the bosonic case are shown, and alternative proofs of known facts are given. Clifford algebras and the Grassmann representation can thus be used to obtain mathematical results regarding coherent fermion states.
We investigate the situation when a normal positive linear unital map on a semifinite von Neumann algebra leaving the trace invariant does not change fixed quantum Renyi's entropy of the density of a normal state. It is also shown that such a map does not change the entropy of any density if and only if it is a Jordan *-isomorphism on the algebra.
We present some results concerning the relative modular operator in semifinite von Neumann algebras. These results allow one to prove some basic formula for trace, to obtain equivalence between Araki's relative entropy and Umegaki's information as well as to derive some formulae for quasi-entropies, and Rényi's relative entropy known in finite dimension.
