Journal of Operator Theory
Volume 67, Issue 1, Winter 2012 pp. 289-298.
An abstract characterization of unital operator spacesAuthors: 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 $\rm CB(A)$ and $\rm CB(\mathcal{M}_*)$ are unital operator spaces, when $A$ is a $C^*$-algebra and $\mathcal{M}$ is a von Neumann algebra. We also show that if $X$ is a normed space with numerical index $1$, then $\rm CB(\max \mathbb{X}; \min \mathbb{X})$ is a unital operator space. Using the idea in our characterization, we consider unital tensor products of unital operator spaces.
Keywords: unital operator algebras
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