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

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.

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