Numerical Radius Norms on Operator Spaces.pdf

a r X i v : m a t h / 0 4 0 4 1 5 3 v 1 [ m a t h .O A ] 7 A p r 2 0 0 4 NUMERICAL RADIUS NORMS ON OPERATOR SPACES TAKASHI ITOH? AND MASARU NAGISA?? 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 oper- ator spaces can be regarded as a subcategory of numerical radius operator spaces. 1. Introduction Let B(H) be the set of all bounded operators on a Hilbert space H, and Hn the n-direct sum of H. We denote by ‖a‖n the operator norm, and wn(a) the numerical radius norm for a ∈ B(Hn) respectively, and identify B(Hn) with the n× n matrix space Mn(B(H)). In [11], Ruan introduced a striking concept of operator spaces. An (abstract) operator space is a complex linear space X together with a sequence of norms On(·) on the n × n matrix space Mn(X) for each n ∈ N, which satisfies the following Ruan’s axioms OI, OII: OI. Om+n ([ x 0 0 y ]) = max{Om(x),On(y)}, OII. On(αxβ) ≤ ‖α‖Om(x)‖β‖ for all x ∈ Mm(X), y ∈ Mn(X) and α ∈ Mn,m(C), β ∈ Mm,n(C). Ruan proved in [11] that if X is an (abstract) operator space, then there is a complete isometry Ψ from X to B(H), that is, ‖[Ψ (xij)]‖n = On([xij ]) for all [xij ] ∈ Mn(X), n ∈ N. In this paper, we introduce an (abstract) numerical radius operator space. We call that X is a numerical radius operator space if a complex 1 2 T. ITOH AND M. NAGISA linear space X admits a sequence of norms Wn(·) on the n× n matrix space Mn(X) for each n ∈ N, which satisfies a couple of conditions WI,WII, where WI is the same as OI, however WII is a