Weak Expectations and the Injective Envelope

TL;DR

This paper investigates the structure of injective envelopes of unital C*-subalgebras within B(H), exploring their images, intersections, and implications for the existence of weak expectations in operator algebras.

Contribution

It introduces a new perspective on the images of injective envelopes and their relation to weak expectations, including the concept of a reflexive cover and order completion.

Findings

The intersection of all images of the injective envelope forms a reflexive cover.

This intersection also acts as a new type of order completion of the algebra.

Results provide insights into conditions for the existence of weak expectations.

Abstract

Given a unital C*-subalgebra of B(H), we study the set of all possible images of its injective envelope that are contained in B(H) and their position relative to the double commutant of the algebra in order to obtain more information about the existence or non-existence of weak expectations. We study the subset of B(H) that is the intersection of all possible images of the injective envelope and show that it is simultaneously a reflexive cover and a new type of order completion of the algebra.