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 MIN(X), which turns out to be a function algebra over the set of extreme points of 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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