Unique Extension Properties for C*-Inclusions

Abstract: Let $\mathcal{A} \subseteq \mathcal{B}$ be a $C^*$-inclusion, i.e., an inclusion of unital $C^*$-algebras with the same unit. Structural properties of the inclusion are often reflected by the fact that certain families of UCP (unital completely positive) maps on $\mathcal{A}$ extend uniquely to UCP maps on $\mathcal{B}$. In particular, depending on the structure of $\mathcal{A} \subseteq \mathcal{B}$, it could be the case that

i. every pure state on $\mathcal{A}$ extends uniquely to a pure state on $\mathcal{B}$ (i.e., $\mathcal{A} \subseteq \mathcal{B}$ has the pure extension property);

ii. a weak* dense set of pure states on $\mathcal{A}$ extend uniquely to pure states on $\mathcal{B}$ (i.e., $\mathcal{A} \subseteq \mathcal{B}$ has the almost extension property);

iii. the identity map $\operatorname{id}:\mathcal{A} \to \mathcal{A}$ extends uniquely to a UCP map $E:\mathcal{B} \to \mathcal{A}$ (i.e., $\mathcal{A} \subseteq \mathcal{B}$ has a unique conditional expectation);

iv. the identity map $\operatorname{id}:\mathcal{A} \to \mathcal{A}$ extends uniquely to a UCP map $\theta:\mathcal{B} \to I(\mathcal{A})$, where $I(\mathcal{A})$ is the injective envelope of $\mathcal{A}$ (i.e., $\mathcal{A} \subseteq \mathcal{B}$ has a unique pseudo-expectation).

In this talk, we explore properties (i)-(iv) above, with a special emphasis on abelian inclusions $C(X) \subseteq C(Y)$ and inclusions $\mathcal{A} \subseteq \mathcal{A} \rtimes_r G$ arising from actions of discrete groups. Applications to determining the simplicity of reduced crossed products are provided.

geometric topologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

( slides | video )

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