## Illinois Journal of Mathematics

### Unique pseudo-expectations for $C^{*}$-inclusions

#### Abstract

Given an inclusion $\mathcal{D}\subseteq\mathcal{C}$ of unital $C^{*}$-algebras (with common unit), a unital completely positive linear map $\Phi$ of $\mathcal{C}$ into the injective envelope $I(\mathcal{D})$ of $\mathcal{D}$ which extends the inclusion of $\mathcal{D}$ into $I(\mathcal{D})$ is a pseudo-expectation. Pseudo-expectations are generalizations of conditional expectations, but with the advantage that they always exist. The set $\operatorname{PsExp}(\mathcal{C},\mathcal{D})$ of all pseudo-expectations is a convex set, and when $\mathcal{D}$ is Abelian, we prove a Krein–Milman type theorem showing that $\operatorname{PsExp}(\mathcal{C},\mathcal{D})$ can be recovered from its set of extreme points. In general, $\operatorname{PsExp}(\mathcal{C},\mathcal{D})$ is not a singleton. However, there are large and natural classes of inclusions (e.g., when $\mathcal{D}$ is a regular MASA in $\mathcal{C}$) such that there is a unique pseudo-expectation. Uniqueness of the pseudo-expectation typically implies interesting structural properties for the inclusion. For general inclusions of $C^{*}$-algebras with $\mathcal{D}$ Abelian, we give a characterization of the unique pseudo-expectation property in terms of order structure; and when $\mathcal{C}$ is Abelian, we are able to give a topological description of the unique pseudo-expectation property.

As applications, we show that if an inclusion $\mathcal{D}\subseteq\mathcal{C}$ has a unique pseudo-expectation $\Phi$ which is also faithful, then the$C^{*}$-envelope of any operator space $\mathcal{X}$ with $\mathcal{D}\subseteq\mathcal{X}\subseteq\mathcal{C}$ is the $C^{*}$-subalgebra of $\mathcal{C}$ generated by $\mathcal{X}$; we also show that for many interesting classes of $C^{*}$-inclusions, having a faithful unique pseudo-expectation implies that $\mathcal{D}$ norms $\mathcal{C}$, although this is not true in general.

#### Article information

Source
Illinois J. Math., Volume 59, Number 2 (2015), 449-483.

Dates
Revised: 23 December 2015
First available in Project Euclid: 5 May 2016

https://projecteuclid.org/euclid.ijm/1462450709

Digital Object Identifier
doi:10.1215/ijm/1462450709

Mathematical Reviews number (MathSciNet)
MR3499520

Zentralblatt MATH identifier
1351.46056

#### Citation

Pitts, David R.; Zarikian, Vrej. Unique pseudo-expectations for $C^{*}$-inclusions. Illinois J. Math. 59 (2015), no. 2, 449--483. doi:10.1215/ijm/1462450709. https://projecteuclid.org/euclid.ijm/1462450709

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