In a 1991 paper, R. Mercer asserted that a Cartan bimodule isomorphism
between Cartan bimodule algebras A_1 and A_2 extends uniquely to a normal
*-isomorphism of the von Neumann algebras generated by A_1 and A_2 [13,
Corollary 4.3]. Mercer's argument relied upon the Spectral Theorem for
Bimodules of Muhly, Saito and Solel [15, Theorem 2.5]. Unfortunately, the
arguments in the literature supporting [15, Theorem 2.5] contain gaps, and
hence Mercer's proof is incomplete.
In this paper, we use the outline in [16, Remark 2.17] to give a proof of
Mercer's Theorem under the additional hypothesis that the given Cartan bimodule
isomorphism is weak-* continuous. Unlike the arguments contained in [13, 15],
we avoid the use of the Feldman-Moore machinery from [8]; as a consequence, our
proof does not require the von Neumann algebras generated by the algebras A_i
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 [1].
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
parametrize 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 weak-* 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.Comment: 21 pages, paper is a completely reworked and expanded version of an
earlier preprint with a similar titl
