Isometric isomorphisms of Cartan bimodule algebras

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This paper is dedicated to proving a single result: that isometric isomorphisms of Cartan bimodule algebras can be extended to ∗-isomorphisms of the generated von Neumann algebras. If M is a von Neumann algebra with Cartan subalgebra A , then a (Cartan) bimodule algebra is simple a σ-weakly closed subalgebra S of M which contains A ; S generates M if there is no proper von Neumann subalgebra of M containing S . The key idea is that a theorem of Muhly, Saito, and Solel concerning isomorphisms of maximal subdiagonal algebras can be extended to this much wider class of algebras if we restrict our attention to isometric isomorphisms. Although the algebras are not assumed to be hyperfinite, a finite-dimensional result of Davidson and Power lies at the leart of the proof. What makes the use of finite-dimensional techniques possible is the existence of “sufficiently many” finite subequivalence relations of an arbitrary countable measurable equivalence relation.