Journal of Operator Theory

Volume 67, Issue 1, Winter 2012 pp. 289-298.

An abstract characterization of unital operator spaces

Authors: Xu-Jian Huang (1), Chi-Keung Ng (2)
Author institution: (1) Department of Mathematics, Tianjin University of Technology, Tianjin 300384, China
(2) Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China

Summary: In this article, we characterize the ``identity'' of an operator space through an analogue of the abstract numerical radius. From this, we give a simple proof of the fact that quotients of unital operator spaces by complete $M$-ideals are unital. Moreover, we show that both $\CB(A)$ and $\CB(\CM_*)$ are unital operator spaces, when $A$ is a $C^*$-algebra and $\CM$ is a von Neumann algebra. We also show that if $X$ is a normed space with numerical index $1$, then $\CB(\max \OX; \min \OX)$ is a unital operator space. Using the idea in our characterization, we consider unital tensor products of unital operator spaces.

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