# Numerical Radius Norms on Operator Spaces

Research paper by **Takashi Itoh, Masaru Nagisa**

Indexed on: **07 Apr '04**Published on: **07 Apr '04**Published in: **Mathematics - Operator Algebras**

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#### 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.