Ring isomorphisms of Murray–von Neumann algebras
Introduction
Let be a von Neumann algebra and let (respectively, ) be a ⁎-algebra of all measurable (respectively, locally measurable) operators with respect to .
In the paper [13] M. Mori characterized lattice isomorphisms between projection lattices and of arbitrary von Neumann algebras and , respectively, by means of ring isomorphisms between the algebras and . In this connection he investigated the following problem.
Question 1.1 Let be von Neumann algebras. What is the general form of ring isomorphisms from onto ?
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 onto has the form where Ψ is a real ⁎-isomorphism from onto and 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 is a finite von Neumann algebra, then (see [12]). If the von Neumann algebra is abelian (i.e. of type I1) then it is ⁎-isomorphic to the algebra of all (classes of equivalence of) essentially bounded measurable complex functions on a measure space and therefore, is the algebra of all measurable complex functions on . 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 (which is isomorphic to ) admits discontinuous algebra automorphisms which identically act on the Boolean algebra of characteristic functions of measurable subsets of the interval .
The following consideration shows that also for the type In case, , 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 be a von Neumann algebra of type In, with the center . Then is ⁎-isomorphic to the algebra of all matrices over (cf. [17, Theorem 2.3.3]). Moreover the algebra is ⁎-isomorphic to the algebra , where is the center of (see [1, Proposition 1.5]). For an arbitrary von Neumann algebra of type In each algebra automorphism Φ of can be represented in the form where is an invertible element and is an extension of a ⁎-automorphism Ψ of the center .
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 and are (say, approximately finite dimensional) II1 factors. Note that cannot have a Banach algebra structure because of the fact that an element of 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 and be type II1 von Neumann algebras. Then any ring isomorphism from onto is a real algebra isomorphism.
Theorem 1.3 Let and be type II1 von Neumann algebras. Then any ring isomorphism from onto 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 and be von Neumann algebras of type II1. Suppose that is a ring isomorphism. Then there exist an invertible element and a real ⁎-isomorphism (which extends to a real ⁎-isomorphism from onto ) such that for all .
Corollary 1.5 Let and be von Neumann algebras of type II1. The projection lattices and are lattice isomorphic, if and only if the von Neumann algebras and are real ⁎-isomorphic (or equivalently, and are Jordan ⁎-isomorphic).
Section snippets
Various isomorphisms of ⁎-algebras
For ⁎-algebras and , a (not necessarily linear) bijection is called
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a ring isomorphism if it is additive and multiplicative;
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a real algebra isomorphism if it is a real-linear ring isomorphism;
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an algebra isomorphism if it is a complex-linear ring isomorphism;
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a real ⁎-isomorphism if it is a real algebra isomorphism and satisfies for all ;
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a ⁎-isomorphism if it is a complex-linear real ⁎-isomorphism.
von Neumann algebras
Let H be a Hilbert space, 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 onto which is not a real algebra isomorphism.
Let be the system of matrix units as in Lemma 2.1. Put and for each take the unitary In the proofs of real-linearity and automatic continuity of ring isomorphisms between the algebras and 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 onto 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 with the following property: for any projection there exists a sequence in such that and .
Proof Consider the separating space of Φ which is defined as follows
General form of ring isomorphisms
In this Section we shall prove Theorem 1.4 which is the main result of the paper.
Let and be arbitrary type II1 von Neumann algebras with faithful normal finite traces and , respectively and let be a ring isomorphism, which is a continuous real algebra isomorphism according to Theorem 1.2, Theorem 1.3.
Lemma 5.1 Let be a projection. Suppose that and , where denotes the support of an element a. Then
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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