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Summer 2015 Unique pseudo-expectations for $C^{*}$-inclusions
David R. Pitts, Vrej Zarikian
Illinois J. Math. 59(2): 449-483 (Summer 2015). DOI: 10.1215/ijm/1462450709


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.


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David R. Pitts. Vrej Zarikian. "Unique pseudo-expectations for $C^{*}$-inclusions." Illinois J. Math. 59 (2) 449 - 483, Summer 2015.


Received: 12 August 2015; Revised: 23 December 2015; Published: Summer 2015
First available in Project Euclid: 5 May 2016

zbMATH: 1351.46056
MathSciNet: MR3499520
Digital Object Identifier: 10.1215/ijm/1462450709

Primary: 46L05, 46L07, 46L10
Secondary: 46M10

Rights: Copyright © 2015 University of Illinois at Urbana-Champaign


Vol.59 • No. 2 • Summer 2015
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