Analysis Seminar

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