Ring Isomorphisms of $\ast$-Subalgebras of Murray–von Neumann Factors

Abstract

The present paper is devoted to study of ring isomorphisms of $\ast$-subalgebras of Murray–von Neumann factors. Let $\mathcal{M},$ $\mathcal{N}$ be von Neumann factors of type II${}_{1},$ and let $S(\mathcal{M}),$ $S(\mathcal{N})$ be the $\ast$-algebras of all measurable operators affiliated with $\mathcal{M}$ and $\mathcal{N},$ respectively. Suppose that $\mathcal{A}\subset S(\mathcal{M}),$ $\mathcal{B}\subset S(\mathcal{N})$ are their $\ast$-subalgebras such that $\mathcal{M}\subset\mathcal{A},$ $\mathcal{N}\subset\mathcal{B}$. We prove that for every ring isomorphism $\Phi:\mathcal{A}\to\mathcal{B}$ there exist a positive invertible element $a\in\mathcal{B}$ with $a^{-1}\in\mathcal{B}$ and a real $\ast$-isomorphism $\Psi:\mathcal{M}\to\mathcal{N}$ (which extends to a real $\ast$-isomorphism from $\mathcal{A}$ onto $\mathcal{B}$) such that $\Phi(x)=a\Psi(x)a^{-1}$ for all $x\in\mathcal{A}$. In particular, $\Phi$ is real-linear and continuous in the measure topology. In particular, noncommutative Arens algebras and noncommutative $L_{log}$-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 $S(\mathcal{M}),$ which shows that the condition $\mathcal{M}\subset\mathcal{A}$ is essential in the above mentioned result.