**
David
R.
Pitts,
University
of
Nebraska-Lincoln**

"Cartan Triples"

Cartan MASAs in von Neumann algebras have been well-studied since the pioneering work of Feldman and Moore in the 1970's. The presence of a Cartan MASA in a a given von Neumann algebra $\mathcal{M}$ is useful for understanding the structure of $\mathcal{M}$. Cartan MASAs arise when applying the group measure space construction with a countable group $\Gamma$ acting essentially freely on the measure space $(X,\mu)$.

Cartan triples generalize Cartan MASAs by relaxing the requirement that the subalgebra be a MASA. Specifically, a Cartan triple $(\mathcal{M},\mathcal{N},\mathcal{D})$ consists of three von Nuemann algebras, $\mathcal{M}\supseteq \mathcal{N}\supseteq \mathcal{D}$ with common unit, such that: $\mathcal{D}$ is abelian, $\mathcal{N}=\mathcal{M}\cap \mathcal{D}'$, there exists a faithful normal conditional expectation $E:\mathcal{M}\rightarrow \mathcal{N}$ and the set of unitaries in $\mathcal{M}$ normalizing $\mathcal{D}$ has weak-$*$ dense span in $\mathcal{M}$. Cartan triples arise in a variety of contexts, notably in the group measure space construction where the group $\Gamma$ is no longer required to act essentially freely.

I
will
discuss
how
(isomorphism
classes
of)
Cartan
tripless
are
in
bijective
correspondence
with
(equivalence
classes
of)
extensions
of
inverse
semigroups,

\[\mathcal{P}\hookrightarrow
\mathcal{G}
\overset{q}{\twoheadrightarrow}\mathcal{S},\]

where
$\mathcal{S}$
is
a
Cartan
inverse
monoid
and
$\mathcal{P}$
belongs
to
a
certain
class
of
Clifford
inverse
monoids.
I
will
also
discuss
a
spectral
theorem
for
Bures-closed
$\mathcal{N}$-bimodules
and
how
it
leads
to
a
form
of
Aoi's
theorem
appropriate
for
Cartan
triples.

This work extends previous joint work with Allan Donsig and Adam Fuller from the setting of Cartan MASAs to the setting of Cartan triples, and is again joint with Donsig and Fuller.

MC 5417