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An abstract characterization of unital operator spaces
- Publication Year :
- 2008
-
Abstract
- In this article, we give an abstract characterization of the ``identity'' of an operator space $V$ by looking at a quantity $n_{cb}(V,u)$ which is defined in analogue to a well-known quantity in Banach space theory. More precisely, we show that there exists a complete isometry from $V$ to some $\mathcal{L}(H)$ sending $u$ to ${\rm id}_H$ if and only if $n_{cb}(V,u) =1$. We will use it to give an abstract characterization of operator systems. Moreover, we will show that if $V$ is a unital operator space and $W$ is a proper complete $M$-ideal, then $V/W$ is also a unital operator space. As a consequece, the quotient of an operator system by a proper complete $M$-ideal is again an operator system. In the appendix, we will also give an abstract characterisation of ``non-unital operator systems'' using an idea arose from the definition of $n_{cb}(V,u)$.<br />Comment: Some remarks were added
- Subjects :
- Mathematics - Operator Algebras
Mathematics - Functional Analysis
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.0805.2447
- Document Type :
- Working Paper