A propagation property of free entropy dimension

All generating sets of all property T von Neumann algebras have free entropy dimension $\leq 1$

Suppose $N$ is a diffuse, property T von Neumann algebra and X is an arbitrary finite generating set of selfadjoint elements for N. By using rigidity/deformation arguments applied to representations of N in full matrix algebras, we deduce that the microstate spaces of X are asymptotically discrete up to unitary conjugacy. We use this description to show that the free entropy dimension of X, $δ_0(X)$, is less than or equal to 1. It follows that when N embeds into the ultraproduct of the hyperfinite $\mathrm{II}_1$-factor, then $δ_0(X)=1$ and otherwise, $δ_0(X)=-\infinity$. This generalizes the …

Two applications of free entropy

Using Voiculsecu's free entropy, it is shown that the free group factors L(F_n) lack property C of Popa and that they lack finite multiplicity abelian subalgebras. Property C is an assymptotic commutivity property.

量子相对熵的新单调性性质

New monotonicity property of the quantum relative entropy

本文证明量子相对熵的局部不连续跳变在量子信道与操作作用下具有不增加的新单调性。

$\mathrm{II}_1$ 因子的性质 (T) 的超迹与熵间隙刻画

Hypertrace and entropy gap characterizations of property (T) for $\mathrm{II}_1$ factors

本文通过构造包含 $\mathrm{II}_1$ 因子 $M$ 的 von Neumann 代数，利用超迹、Furstenberg 熵和熵间隙给出性质 (T) 与 Haagerup 性质的等价刻画。