Approximate equivalence in von Neumann algebras

Pub Date : 2023-01-01 DOI:10.7153/oam-2023-17-01
Qihui Li, Don Hadwin, Wenjing Liu
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引用次数: 1

Abstract

Suppose $\mathcal{A}$ is a separable unital ASH C*-algebra, $\mathcal{R}$ is a sigma-finite II$_{\infty}$ factor von Neumann algebra, and $\pi,\rho :\mathcal{A}\rightarrow\mathcal{R}$ are unital $\ast$-homomorphisms such that, for every $a\in\mathcal{A}$, the range projections of $\pi\left( a\right) $ and $\rho\left( a\right) $ are Murray von Neuman equivalent in $\mathcal{R}% $. We prove that $\pi$ and $\rho$ are approximately unitarily equivalent modulo $\mathcal{K}_{\mathcal{R}}$, where $\mathcal{K}_{\mathcal{R}}$ is the norm closed ideal generated by the finite projections in $\mathcal{R}$. We also prove a very general result concerning approximate equivalence in arbitrary finite von Neumann algebras.
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冯诺依曼代数中的近似等价
假设$\mathcal{A}$是一个可分的一元ASH C*-代数,$\mathcal{R}$是一个sigma-finite II$_{\ inty}$因子von Neumann代数,$\pi,\rho:\mathcal{A}\右row\mathcal{R}$是一元$\ast$-同态,使得对于\mathcal{A}$中的每一个$ A \ \, $\pi\左(A \右)$和$\rho\左(A \右)$的范围投影在$\mathcal{R}% $中是Murray von Neumann等价的。证明$\pi$和$\rho$是近似一元等价模$\mathcal{K}_{\mathcal{R}}$,其中$\mathcal{K}_{\mathcal{R}}$是由$\mathcal{R}$中的有限投影生成的范数闭理想。我们还证明了关于任意有限冯诺依曼代数近似等价的一个非常一般的结果。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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