Notes on automorphisms of ultrapowers of II_1 factors

不可约嵌入到超幂 II₁ 因子中而无 UCPh 抬升

Elementary embeddings into ultrapower $\mathrm{II}_1$ factors without a UCP lift

本文研究了 II₁ 因子的不可约嵌入问题，证明了某些嵌入无法通过 UCPh 映射序列提升，并探讨了超幂因子里自同构的基数性质。

A remark on the ultrapower algebra of the hyperfinite factor

On page 43 in \cite{Po83} Sorin Popa asked whether the following property holds: \emph{If $ω$ is a free ultrafilter on $\mathbb N$ and $\mathcal R_1\subseteq \mathcal R$ is an irreducible inclusion of hyperfinite II$_1$ factors such that $\mathcal R'\cap \mathcal R^ω\subseteq \mathcal R^ω_1$ does it follows that $\mathcal R_1=\mathcal R$?} In this short note we provide an affirmative answer to this question.

A note on the invariant subspace problem relative to a type ${\rm II}_1$ factor

Let $\M$ be a type ${\rm II}_1$ factor with a faithful normal tracial state $τ$ and let $\M^ω$ be the ultrapower algebra of $\M$. In this paper, we prove that for every operator $T\in \M^ω$, there is a family of projections $\{P_t\}_{0\leq t\leq 1}$ in $\M^ω$ such that $TP_t=P_tTP_t$, $P_s\leq P_t$ if $s\leq t$, and $τ_ω(P_t)=t$. Let $\mathfrak{M}=\{Z \in \M: \text{there is a family of projections} \{P_t\}_{0\leq t\leq 1} \text{in} \M \text{such that} ZP_t=P_tZP_t, P_s\leq P_t \text{if} s\leq t, \text{and} τ(P_t)=t\}$. As an application we show that for every operator $T\in \M$ and $ε>0$, ther…

A note on equicontinuity of families of operators and automorphisms

This note concerns uniform equicontinuity of families of operators on a separable Hilbert space H, and of families of maps on B(H). It is shown that a one parameter group of automorphisms is uniformly equicontinuous if and only if the group of unitaries which implements it is so. A "geometrical" necessary and sufficient condition is given for a family of operators to be uniformly equicontinuous.

关于 Cuntz 代数自同构的注记

A note on the Cuntz algebra automorphisms

本文刻画了 $[n]^2$ 的一类稳定对合，证明了 Brenti 和 Conti 的猜想，从而为任意 $n>1$ 的 Cuntz 代数 $\mathcal{O}_n$ 提供了一个具有 6 个自由度的自同构新族。