论文标题
$ \ ast $ -subalgebras的戒指同构诺伊曼因素
Ring isomorphisms of $\ast$-subalgebras of Murray-von Neumann factors
论文作者
论文摘要
本文致力于研究Murray的$ \ ast $ -subalgebras-von neumann因素的环形同构。令$ \ cm，$ $ \ cn $是II型$ _1，$和$ _1，$ s（\ cm）的Von Neumann因素，$ s（\ cn）$是$ \ cm $ and $ \ cm $和$ \ cn的所有可测量运算符的$ \ ast $ algebras。假设$ \ ca \ subset s（\ cm），$ $ \ cb \ subset s（\ cn）$是他们的$ \ ast $ -subalgebras，以至于$ \ cm \ cm \ subset \ ca，$ $ $ $ \ cn \ cn \ cn \ subset \ cb。 $ a \ in \ cb $带有$ a^{ - 1} \ in \ cb $和一个真正的$ \ ast $ -isomorphism $ψ：\ cm \ to \ cn $（该$ \ ast $ \ ast $ -isomorphism从$ \ ca $ to $ \ ca $ to $ \ cb $ to $ \ cb $） $ x \ in \ ca. $，尤其是$φ$是现实的，并且在度量拓扑中是连续的。特别是，与von Neumann因子II $ _1 $相关的非交换性Arens代数和非交换性$ \ cl_ {log} $ - 代数满足了上述条件，并且主要定理意味着相应的Metrics中其环同构的自动连续性。我们还提供了$ s（\ cm）的$ \ ast $ -subalgebra的示例，$，该$表明条件$ \ cm \ subset \ ca $在上述结果中至关重要。
The present paper is devoted to study of ring isomorphisms of $\ast$-subalgebras of Murray--von Neumann factors. Let $\cM,$ $\cN$ be von Neumann factors of type II$_1,$ and let $S(\cM),$ $S(\cN)$ be the $\ast$-algebras of all measurable operators affiliated with $\cM$ and $ \cN,$ respectively. Suppose that $\cA\subset S(\cM),$ $\cB\subset S(\cN)$ are their $\ast$-subalgebras such that $\cM\subset \cA,$ $\cN\subset \cB.$ We prove that for every ring isomorphism $Φ: \cA \to \cB$ there exist a positive invertible element $a \in \cB$ with $a^{-1}\in \cB$ and a real $\ast$-isomorphism $Ψ: \cM \to \cN$ (which extends to a real $\ast$-isomorphism from $\cA$ onto $\cB$) such that $Φ(x) = aΨ(x)a^{-1}$ for all $x \in \cA.$ In particular, $Φ$ is real-linear and continuous in the measure topology. In particular, noncommutative Arens algebras and noncommutative $\cL_{log}$-algebras associated with von Neumann factors of type II$_1$ satisfy the above conditions and the main Theorem implies the automatic continuity of their ring isomorphisms in the corresponding metrics. We also present an example of a $\ast$-subalgebra in $S(\cM),$ which shows that the condition $\cM\subset \cA$ is essential in the above mentioned result.
