Date of this Version
2016 University of Illinois
Given an inclusion D⊆C of unital C ∗ -algebras (with common unit), a unital completely positive linear map Φ of C into the injective envelope I(D) of D which extends the inclusion of D into I(D) is a pseudo-expectation. Pseudo-expectations are generalizations of conditional expectations, but with the advantage that they always exist. The set PsExp(C,D) of all pseudo-expectations is a convex set, and when D is Abelian, we prove a Krein–Milman type theorem showing that PsExp(C,D) can be recovered from its set of extreme points. In general, PsExp(C,D) is not a singleton. However, there are large and natural classes of inclusions (e.g., when D is a regular MASA in 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 D Abelian, we give a characterization of the unique pseudoexpectation property in terms of order structure; and when 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 D ⊆ C has a unique pseudo-expectation Φ which is also faithful, then the C ∗ -envelope of any operator space X with D⊆X ⊆C is the C ∗ - subalgebra of C generated by X; we also show that for many interesting classes of C ∗ -inclusions, having a faithful unique pseudoexpectation implies that D norms C, although this is not true in general.