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Transactions of the American Mathematical Society

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Weak expectations and the injective envelope

Author: Vern I. Paulsen
Journal: Trans. Amer. Math. Soc. 363 (2011), 4735-4755
MSC (2010): Primary 46L07; Secondary 47L25
Published electronically: April 8, 2011
MathSciNet review: 2806689
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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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Additional Information

Vern I. Paulsen
Affiliation: Department of Mathematics, University of Houston, Houston, Texas 77204-3476

Keywords: Weak expectation, injective
Received by editor(s): July 24, 2008
Received by editor(s) in revised form: August 25, 2009, and September 1, 2009
Published electronically: April 8, 2011
Additional Notes: This research was supported in part by NSF grant DMS-0600191
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.