Let $LS(\mathcal{M})$ be the algebra of locally measurable operators affiliated with a von Neumann algebra $\mathcal{M}$, equipped with an $F$-norm defined via a dimension function and a probability measure. We prove that every bijective linear isometry between $LS(\mathcal{M})$ admits a canonical representation of the form $\Phi(x)=wJ(x)$, where $w$ is a unitary element and $J$ is a Jordan $^*$-isomorphism, which extends classical results such as the Banach--Stone theorem and Kadison's theorem. Under several structural assumptions on the underlying von Neumann algebras (including all type $\mathrm{II}_\infty$ and type $\mathrm{III}$ algebras, and all factors, and algebras with atomless centers), we prove the one-to-one correspondence between the $F$-norm and the pair $(\mu, D)$ of a probability measure and a dimension function, which fails for algebras with atomic centers.
Both the uniform Roe algebras and the quasi-local algebras encode the large scale geometry of metric spaces. Whether these algebras coincide is a question which goes back to Roe and has only been recently solved by Ozawa. We propose a dynamical point of view on this problem. Given a set $X$, every map $h\colon X\to\mat...
We establish a noncommutative version of a familiar Johnson--Maurey--Schechtman--Tzafriri Theorem, by showing that for any $0<p<2$ and a (not necessarily semifinite) von Neumann algebra $\mathcal{M}$ on a separable Hilbert space, if a symmetric quasi-Banach function space $E(0,1) $ containing the function $t\mapsto t^{...
Jing-Hao Huang, M. Junge, F. Sukochev et al.· 1 citation
The primary aim of this paper is to show a linear topological isomorphism between (elements of a wide class {of}) symmetric spaces over the hyperfinite $II_1$ factor $\mathcal{R}$ and certain symmetric operator space over the hyperfinite $II_\infty $ factor $\mathcal{R}\bar{\otimes}\mathcal{L}(H))$. Precisely, we show...
Let $\mathcal{H}$ be a complex Hilbert space and $\mathcal{B}(\mathcal{H})$ be the algebra of all bounded linear operators on $\mathcal{H}$. For $A \in \mathcal{B}(\mathcal{H})$, we refer to the sequence $\{|A^{n}|^{1/n}\}_{n\in\mathbb{N}}$ as the normalized power sequence of $A$. In this article, we study the norm con...
Let $\mathcal{A}$ be a simple unital $C^*$-algebra with real rank zero, nonempty tracial state space $T(\mathcal{A})$, and strict comparison of projections. We prove that every element $T$ in $\mathcal{A}$ with $\tau(T)=0$ for all $\tau\in T(\mathcal{A})$ is a single commutator if one of the following conditions holds:...
For $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ and every separable Banach space $V$, we use an oracle-relative two-sorted finite-extension construction, whose oracle records the local rational structure of $V$, to construct a separable Banach space $X_V$ such that $$ \operatorname{Cal}(X_V)=\mathcal{B}(X_V)/\mathcal{K}(X_...
Rui-Ke Liu, Jie Shen· 0 citations
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