A regular inclusion is a pair $(\mathcal{C},\mathcal{D})$ of unital $C^*$-algebras (with the same unit) where $\mathcal{D}\subseteq \mathcal{C}$, $\mathcal D$ is abelian, and the set $\{v\in\mathcal{C}: v\mathcal{D} v^*\cup v^*\mathcal{D} v\subseteq\mathcal{D}\}$ has dense span in $\mathcal{C}$.

An important and well-studied class of regular inclusions are Cartan inclusions, which were introduced by Renault (building upon work of Kumjian). Renault makes a strong case that Cartan inclusions are the appropriate $C^*$-algebraic variant of a Cartan MASA in a von Neumann algebra. In addition, Cartan inclusions have numerous useful structural properties.

In previous work, I gave necessary and sufficient conditions for a regular inclusion $(\mathcal C, \mathcal D)$ to regularly embed into a Cartan inclusion. However, the Cartan inclusion into which it is embedded need not be closely related to $(\mathcal C, \mathcal D)$. For example, the identity map on $C([0,1])$ is a regular embedding of $(C([0,1]), \mathbb C I)$ into the Cartan inclusion $(C([0,1]), C([0,1]))$.

In this talk, I will address this defect by defining the notion of a Cartan envelope for a regular inclusion $(\mathcal C,\mathcal D)$; the idea is that the Cartan envelope should be the smallest Cartan pair generated by $(\mathcal C, \mathcal D)$. I will discuss the following result.

Theorem. Let $(\mathcal C,\mathcal D)$ be a regular inclusion. The following statements are equivalent.
(1) $(\mathcal C,\mathcal D)$ has a Cartan envelope.
(2) The relative commutant, $\mathcal D^c$, of $\mathcal D$ in $\mathcal C$ is abelian and both the inclusions $(\mathcal C, \mathcal D^c)$ and $(\mathcal D^c, \mathcal D)$ have the ideal intersection property.
(3) $(\mathcal C,\mathcal D)$ has the unique faithful pseudo-expectation property.

Furthermore, when the Cartan envelope $(\mathcal C_{\rm env}, \mathcal D_{\rm env})$ exists, it is unique (up to regular isomorphism) and minimal in the sense that if $(\mathcal C_1,\mathcal D_1)$ is a Cartan pair and $\alpha:(\mathcal C,\mathcal D)\hookrightarrow (\mathcal C_1,\mathcal D_1)$ is a regular embedding, then there is a regular epimorphism $q: (\mathcal C_1,\mathcal D_1)\twoheadrightarrow (\mathcal C_{\rm env}, \mathcal D_{\rm env})$ such that $q\circ\alpha$ is the embedding of $(\mathcal C,\mathcal D)$ into $(\mathcal C_{\rm env}, \mathcal D_{\rm env})$.