 

Jan Cameron, David R. Pitts, and Vrej Zarikian
Bimodules over Cartan MASAs in von Neumann algebras, norming algebras, and Mercer's Theorem view print


Published: 
August 7, 2013 
Keywords: 
Norming algebra, Cartan MASA, C*diagonal 
Subject: 
47L30, 46L10, 46L07 


Abstract
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} (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 σweakly continuous. Unlike
the arguments contained in the abovementioned papers of Mercer and
MuhlySaitoSolel,
we avoid the use of the machinery in
FeldmanMoore, 1977; 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, 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 Buresclosed 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 σweakly 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.


Acknowledgements
Zarikian was partially supported by Nebraska IMMERSE.


Author information
Jan Cameron:
Dept. of Mathematics, Vassar College, Poughkeepsie, NY 12604
jacameron@vassar.edu
David R. Pitts:
Dept. of Mathematics, University of NebraskaLincoln, Lincoln, NE 685880130
dpitts2@math.unl.edu
Vrej Zarikian:
Dept. of Mathematics, U. S. Naval Academy, Annapolis, MD 21402
zarikian@usna.edu

