Numerical Radius Norms on Operator Spaces

Abstract. We introduce a numerical radius operator space (X,Wn). The conditions to be a numerical radius operator space are weaker than the Ruan’s axiom for an operator space (X,On). Let w(·) be the numerical radius norm on B(H). It is shown that if X admits a norm Wn(·) on the matrix space Mn(X) which satisfies the conditions, then there is a complete isometry, in the sense of the norms Wn(·) and wn(·), from (X,Wn) into (B(H), wn). We study the relationship between the operator space (X,On) and the numerical radius operator space (X,Wn). The category of operator spaces can be regarded as a subcategory of numerical radius operator spaces.