Numerical Radius Norms on Operator Spaces

Abstract

We introduce a numerical radius operator space $(X,{\mathcal{W}}_n)$. The conditions to be a numerical radius operator space are weaker than the Ruan's axiom for an operator space $(X,{\mathcal{O}}_n)$. Let $w(\cdot )$ be the numerical radius norm on $\mathbb{B}(\mathcal{H})$. It is shown that if $X$ admits a norm ${\mathcal{W}}_n(\cdot )$ on the matrix space ${\mathbb{M}}_n(X)$ which satisfies the conditions, then there is a complete isometry, in the sense of the norms ${\mathcal{W}}_n(\cdot )$ and $w_n(\cdot )$, from $(X,{\mathcal{W}}_n)$ into $(\mathbb{B}(\mathcal{H}),w_n)$. We study the relationship between the operator space $(X,{\mathcal{O}}_n)$ and the numerical radius operator space $(X,{\mathcal{W}}_n)$. The category of operator spaces can be regarded as a subcategory of numerical radius operator spaces.

18 pages

References in corpus (1)

• The numerical radius Haagerup norm and Hilbert space square factorizations