ACTA issues

## Numerical radius norm for bounded module maps and Schur multipliers

Takashi Itoh, Masaru Nagisa

Acta Sci. Math. (Szeged) 70:1-2(2004), 195-211
5808/2009

 Abstract. Let ${\cal A}$ be a unital, norm-closed subalgebra of the bounded operators ${\msbm B}({\cal H})$ on a Hilbert space ${\cal H}$ and $T$ a normal left ${\cal A}^*$-, right ${\cal A}$-module completely bounded map of ${\msbm B}({\cal H})$. For the numerical radius norm $w(\cdot )$ on ${\msbm B}({\cal H})$, we let $\|T\|_w = \sup\{w(T(x)) | w(x) \le1 \}$ and $\|T\|_{wcb}= \sup_n \|T \otimes\mathop{\rm id}_n \|_w$. It is shown that there exist $t=(t_{ij})\in{\msbm B}(\ell ^2(I))$ and elements $v_i$ ($i \in I$) of the commutant of ${\cal A}$ such that $\|t\|=\|T\|_{wcb}$, $\sum_{i\in I}v_i^*v_i \le1$, and $T(x) = \sum_{i,j\in I}v_i^*t_{ij}xv_j$ $(x\in{\msbm B}({\cal H}))$. As an application, we generalize Ando--Okubo's theorem for Schur multipliers on ${\msbm B}({\cal H})$. AMS Subject Classification (1991): 46L05, 46L07, 47C15 Keyword(s): Completely bounded, Completely positive, Schur multipliers, Numerical radius norm, Operator systems Received May 12, 2003, and in revised form October 29, 2003. (Registered under 5808/2009.)