Abstract
Abstract Generalizing von Neumann’s result on type II $_1$ von Neumann algebras, I characterise lattice isomorphisms between projection lattices of arbitrary von Neumann algebras by means of ring isomorphisms between the algebras of locally measurable operators. Moreover, I give a complete description of ring isomorphisms of locally measurable operator algebras when the von Neumann algebras are without type II direct summands.
Highlights
In the case of type I∞ factors, we can make use of a result from [5]
There exists a real ∗-isomorphism Ψ : → that extends Φ. Each of these results implies that lattice isomorphisms between projection lattices are closely related to ring isomorphisms
It is natural to imagine that we can obtain a similar result for arbitrary lattice isomorphisms in the general setting of von Neumann algebras
Summary
We use the symbol ∼ to mean the Murray–von Neumann equivalence relation on P( ). For , ∈ P( ), ⊥ means that and are orthogonal. The symbol Z( ) = { ∈ | = for all ∈ } means the center of. It is well known that every von Neumann algebra without finite type I direct summands has order for any ∈ N [9, Lemma 6.5.6]. If has order ∈ N, it can be identified with the algebra M ( ˆ ) of × matrices with entries in some von Neumann algebra
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