Ring isomorphisms of $\ast$-subalgebras of Murray-von Neumann factors

TL;DR

This paper characterizes ring isomorphisms between certain *-subalgebras of Murray-von Neumann factors, showing they are essentially conjugations by invertible elements combined with *-isomorphisms of the factors, ensuring automatic continuity.

Contribution

It establishes that ring isomorphisms of *-subalgebras containing the factors are implemented by conjugation with invertible elements and extend to *-isomorphisms of the factors, with automatic continuity.

Findings

Ring isomorphisms are implemented by conjugation with invertible elements.

Such isomorphisms extend to *-isomorphisms of the underlying factors.

They are automatically continuous in the measure topology.

Abstract

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 $\Phi: \cA \to \cB$ there exist a positive invertible element $a \in \cB$ with $a^{-1}\in \cB$ and a real $\ast$-isomorphism $\Psi: \cM \to \cN$ (which extends to a real $\ast$-isomorphism from $\cA$ onto $\cB$) such that $\Phi(x) = a\Psi(x)a^{-1}$ for all $x \in \cA.$ In particular, $\Phi$ 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.