Ring isomorphisms of Murray–von Neumann algebras

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Abstract

We give a complete description of ring isomorphisms between algebras of measurable operators affiliated with von Neumann algebras of type II1.

Introduction

Let M be a von Neumann algebra and let S(M) (respectively, LS(M)) be a ⁎-algebra of all measurable (respectively, locally measurable) operators with respect to M.

In the paper [13] M. Mori characterized lattice isomorphisms between projection lattices P(M) and P(N) of arbitrary von Neumann algebras M and N, respectively, by means of ring isomorphisms between the algebras LS(M) and LS(N). In this connection he investigated the following problem.

Question 1.1

Let M,N be von Neumann algebras. What is the general form of ring isomorphisms from LS(M) onto LS(N)?

In [13, Theorem B] Mori himself gave an answer to the above Question in the case of von Neumann algebras of type I and III. Namely, any ring isomorphism Φ from LS(M) onto LS(N) has the formΦ(x)=yΨ(x)y1,xLS(M), where Ψ is a real ⁎-isomorphism from LS(M) onto LS(N) and yLS(N) is an invertible element. Note that in the case where Φ is an algebraic isomorphism of type I von Neumann algebras, the above presentation was obtained in [2].

If M is a finite von Neumann algebra, then LS(M)=S(M) (see [12]). If the von Neumann algebra M is abelian (i.e. of type I1) then it is ⁎-isomorphic to the algebra L(Ω,Σ,μ) of all (classes of equivalence of) essentially bounded measurable complex functions on a measure space (Ω,Σ,μ) and therefore, S(M)S(Ω,Σ,μ) is the algebra of all measurable complex functions on (Ω,Σ,μ). A.G. Kusraev [10] by means of Boolean-valued analysis establishes necessary and sufficient conditions for existence of discontinuous non trivial algebra automorphisms on extended complete complex f-algebras. In particular, he has proved that the algebra S[0,1] (which is isomorphic to LS(L[0,1])=S(L[0,1])) admits discontinuous algebra automorphisms which identically act on the Boolean algebra P(L[0,1]) of characteristic functions of measurable subsets of the interval [0,1].

The following consideration shows that also for the type In case, 1<n<, ring isomorphisms may be discontinuous in general (see for details [2]) and therefore the representation from [13, Theorem B] is not valid for this case.

Let M be a von Neumann algebra of type In, 1<n< with the center Z(M). Then M is ⁎-isomorphic to the algebra Mn(Z(M)) of all n×n matrices over Z(M) (cf. [17, Theorem 2.3.3]). Moreover the algebra S(M) is ⁎-isomorphic to the algebra Mn(Z(S(M))), where Z(S(M))=S(Z(M)) is the center of S(M) (see [1, Proposition 1.5]). For an arbitrary von Neumann algebra M of type In each algebra automorphism Φ of S(M) can be represented in the formΦ(x)=aΨ(x)a1,xS(M), where aS(M) is an invertible element and Ψ is an extension of a ⁎-automorphism Ψ of the center S(Z(M)).

In [13] the author conjectured that the representation of ring isomorphisms, mentioned above for type I and III cases holds also for type II von Neumann algebras. At the end of the paper M. Mori wrote that “The author does not know whether or not such a Φ is automatically real-linear even in the case M and N are (say, approximately finite dimensional) II1 factors. Note that LS(M) cannot have a Banach algebra structure because of the fact that an element of LS(M) can have an empty or dense spectral set. Hence it seems to be difficult to make use of automatic continuity results on algebra isomorphisms as in [4]”.

In the present paper we give an answer to the Question 1.1 for type II1 von Neumann algebras. The paper is organized as follows.

In Section 2 we give definitions of various kinds of isomorphisms between ⁎-algebras and also some preliminaries from the theory of measurable operators affiliated with von Neumann algebras.

In order to prove the main result of the present paper, in Sections 3 and 4 we show automatic real-linearity and automatic continuity of ring isomorphisms between algebras of measurable operators affiliated with von Neumann algebras of type II1. Namely, we prove the following two theorems.

Theorem 1.2

Let M and N be type II1 von Neumann algebras. Then any ring isomorphism from S(M) onto S(N) is a real algebra isomorphism.

Theorem 1.3

Let M and N be type II1 von Neumann algebras. Then any ring isomorphism from S(M) onto S(N) is continuous in the local measure topology.

In Section 5 the following main result confirms the Conjecture 5.1 in [13] and answers the above Question 1.1 for the type II1 case.

Theorem 1.4

Let M and N be von Neumann algebras of type II1. Suppose that Φ:S(M)S(N) is a ring isomorphism. Then there exist an invertible element aS(N) and a real-isomorphism Ψ:MN (which extends to a real-isomorphism from S(M) onto S(N)) such that Φ(x)=aΨ(x)a1 for all xS(M).

Corollary 1.5

Let M and N be von Neumann algebras of type II1. The projection lattices P(M) and P(N) are lattice isomorphic, if and only if the von Neumann algebras M and N are real-isomorphic (or equivalently, M and N are Jordan-isomorphic).

Section snippets

Various isomorphisms of ⁎-algebras

For ⁎-algebras A and B, a (not necessarily linear) bijection Φ:AB is called

  • a ring isomorphism if it is additive and multiplicative;

  • a real algebra isomorphism if it is a real-linear ring isomorphism;

  • an algebra isomorphism if it is a complex-linear ring isomorphism;

  • a real ⁎-isomorphism if it is a real algebra isomorphism and satisfies Φ(x)=Φ(x) for all xA;

  • a ⁎-isomorphism if it is a complex-linear real ⁎-isomorphism.

von Neumann algebras

Let H be a Hilbert space, B(H) be the ⁎-algebra of all bounded linear

Real-linearity of ring isomorphisms

In this section we prove Theorem 1.2 in a series of Lemmas.

Suppose the contrary and assume that Φ is a ring isomorphism from S(M) onto S(N) which is not a real algebra isomorphism.

Let {eij(n):n=0,1,,i,j=1,,2n} be the system of matrix units as in Lemma 2.1. Put u0=1 and for each n1 take the unitaryun=i=12n1ei,i+1(n)+e2n,1(n)N. In the proofs of real-linearity and automatic continuity of ring isomorphisms between the algebras S(M) and S(N) we will essentially use this family of unitaries.

Continuity of ring isomorphisms in the measure topology

In this Section we assume that Φ is a real algebra isomorphism from S(M) onto S(N) which is discontinuous in the measure topology. In order to come to a contradiction we should slightly modify the proof from the previous Section.

Lemma 4.1

There exists a non-zero projection zP(Z(N)) with the following property: for any projection eP(zN) there exists a sequence {xn} in S(M) such that xntτM0 and Φ(xn)tτNe.

Proof

Consider the separating space of Φ which is defined as followsS(Φ)={yS(N):{xn}S(M),xnS(N)0,Φ(x

General form of ring isomorphisms

In this Section we shall prove Theorem 1.4 which is the main result of the paper.

Let M and N be arbitrary type II1 von Neumann algebras with faithful normal finite traces τM and τN, respectively and let Φ:MN be a ring isomorphism, which is a continuous real algebra isomorphism according to Theorem 1.2, Theorem 1.3.

Lemma 5.1

Let pNP(N) be a projection. Suppose that xM and qM=1s(Φ1(Φ(x)))s(Φ1(pN)), where s(a) denotes the support of an element a. Then(pN+qN)Φ(x+y)Φ(x+y)(pN+qN)=pNΦ(x)Φ(x)pN+qNΦ(y)Φ

Acknowledgement

We are indebted to the Referee for very valuable suggestions and comments, which helped us to significantly improve the exposition.

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