Ring Isomorphisms of *-Subalgebras of Murray–von Neumann Factors

Abstract: The present paper is devoted to study of ring isomorphisms of *-subalgebras of Murray–von Neumann factors. Let M,N be von Neumann factors of type II₁ and let S(M),S(N) be the *-algebras of all measurable operators affiliated with M and N respectively. Suppose that A S(M), B S(N) are their *-subalgebras such that M A S(M), B S(N). We prove that for every ring isomorphism Φ :A→B there exist a positive invertible element (Formula Presented.) and a real *-isomorphism Φ :M→N (which extends to a real *-isomorphism from Ψ :M→N such that Φ(x) = aΨ(x)a⁻¹ for all x A.. In particular, Φ is real-linear and continuous in the measure topology. In particular, noncommutative Arens algebras and noncommutative (Formula Presented.)-algebras associated with von Neumann factors of type II₁ 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 *-subalgebra in S(M) which shows that the condition M A is essential in the above mentioned result.