Bimodules over Cartan MASAs in von Neumann algebras, norming algebras, and Mercer's Theorem

摘要

In a 1991 paper, R. Mercer asserted that a Cartan bimodule isomorphism between Cartan bimodule algebras A1 and A2 extends uniquely to a normal *-isomorphism of the von Neumann algebras generated by A1 and A2 (Corollary 4.3 of Mercer, 1991). Mercer's argument relied upon the Spectral Theorem for Bimodules of Muhly, Saito and Solel, 1988 (Theorem 2.5, there). Unfortunately, the arguments in the literature supporting their Theorem 2.5 contain gaps, and hence Mercer's proof is incomplete. In this paper, we use the outline in Pitts, 2008, Remark 2.17, to give a proof of Mercer's Theorem under the additional hypothesis that the given Cartan bimodule isomorphism is σ-weakly continuous. Unlike the arguments contained in the abovementioned papers of Mercer and Muhly-Saito-Solel, we avoid the use of the machinery in Feldman-Moore, 1977; as a consequence, our proof does not require the von Neumann algebras generated by the algebras Ai to have separable preduals. This point of view also yields some insights on the von Neumann subalgebras of a Cartan pair (M,D), for instance, a strengthening of a result of Aoi, 2003. We also examine the relationship between various topologies on a von Neumann algebra M with a Cartan MASA D. This provides the necessary tools to parameterize the family of Bures-closed bimodules over a Cartan MASA in terms of projections in a certain abelian von Neumann algebra; this result may be viewed as a weaker form of the Spectral Theorem for Bimodules, and is a key ingredient in the proof of our version of Mercer's Theorem. Our results lead to a notion of spectral synthesis for σ-weakly closed bimodules appropriate to our context, and we show that any von Neumann subalgebra of M which contains D is synthetic. We observe that a result of Sinclair and Smith shows that any Cartan MASA in a von Neumann algebra is norming in the sense of Pop, Sinclair and Smith.