Weak expectations and the injective envelope

Paulsen, Vern

doi:10.1090/S0002-9947-2011-05203-7

Weak expectations and the injective envelope
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by Vern I. Paulsen

Trans. Amer. Math. Soc. 363 (2011), 4735-4755

DOI: https://doi.org/10.1090/S0002-9947-2011-05203-7

Published electronically: April 8, 2011

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Abstract:

Given a unital $C^*$-subalgebra $\mathcal A \subseteq B(\mathcal H),$ we study the set of all possible images of the injective envelope $I(\mathcal A)$ of $\mathcal A$ that are contained in $B(\mathcal H)$ and their position relative to the double commutant of the algebra in order to develop more information about the existence or non-existence of weak expectations. We study the set of all elements of $B(\mathcal H)$ that are fixed by all completely positive maps that fix $\mathcal A.$ We also introduce a new category, such that the injective envelope of $\mathcal A$ in the new category is always contained in the double commutant of $\mathcal A.$ We study the relationship between these two injective envelopes and the existence of weak expectations.

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