Pseudo-Cartan Inclusions and their Cartan Envelopes

Abstract: I will discuss the class of pseudo-Cartan inclusions, which are a class of regular inclusions of $C^*$-algebras $\mathcal D\subseteq \mathcal C$ where $\mathcal D$ is abelian. This class includes several previously studied classes such as: Cartan inclusions, weak Cartan inclusions and virtual Cartan inclusions.

The class of pseudo-Cartan inclusions coincides with the class of regular inclusions having a Cartan envelope. Roughly speaking, a Cartan envelope for a regular inclusion is a minimal Cartan inclusion into which the inclusion regularly embeds.

Pseudo-Cartan inclusions and their Cartan envelopes have desirable properties: for example, they behave well under suitable inductive limits and under minimal tensor products. Time permitting, I will describe some applications. Here is a sample Application: Suppose for $i=1,2$, $(\mathcal C_i,\mathcal D_i)$ are pseudo-Cartan inclusions and $\mathcal A_i$ are intermediate Banach algebras, \[\mathcal D_i\subseteq \mathcal A_i\subseteq \mathcal C_i.\] If $\theta: \mathcal A_1\rightarrow \mathcal A_2$ is an isometric isomorphism, then $\theta$ uniquely extends to a $*$-isomorphism of the $C^*$-subalgebras of $\mathcal C_i$ generated by $\mathcal A_i$, \[\tilde\theta: C^*(\mathcal A_1)\rightarrow C^*(\mathcal A_2).\]

geometric topologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

( paper | slides | video )

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