Journal of Operator Theory

Volume 33, Issue 2, Spring 1995 pp. 327-351.

Representations of operator spaces

Authors: Chun Zhang
Author institution:Department of Mathematics, The University of Houston, Houston, TX 77204, U.S.A.

Summary: Let $V$ be any abstract operator space. We represent it completely isometrically into some $\mathcal B (H)$ in various ways, then examine the different $C^*$-algebras and different operator systems it generates. In particular, we construct two $C^*$-envelopes of an operator space. Using the off-diagonal representation $v \mapsto \left[ {\begin{array}{*{20}c}
0 & v \\
0 & 0 \\
\end{array}} \right]$, from any operator space we are able to build two $C^*$-algebras which are Morita equivalent $C^*$-algebras. As an application, we compute the $C^*$-envelope of $\rm MIN(X)$, which turns out to be a function algebra over the set of extreme points of $\rm Ball(X')$ modulo the action of the unit circle. Finally, we introduce a partial ordering on the operator systems spanned by an operator space. We show that there is a maximal element with respect to this ordering.

Keywords: Operator space, operator system, $C^*$-envelope, $C^*$-algebra.

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