2008/05/16 by Xu-Jian Huang, Huang, Xu-Jian, Chi-Keung Ng +1
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #math.FA #math.OA
paper · pdf · doi:10.48550/arxiv.0805.2447
Some remarks were added
arxiv created 2008/05/27 · arxiv updated 2009/12/01
In this article, we give an abstract characterization of the ``identity'' of an operator space V by looking at a quantity ncb(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 L(H) sending u to \rm idH if and only if ncb(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 ncb(V,u).