A formula for the numerical range of an elementary operator on C*-algebra

Mohamed Barraa

paper

A formula for the numerical range of an elementary operator on C*-algebra

arXiv:1504.04569

Abstract

Let $\A$ be a $C^{*} -$ algebra with unit element $1$ and unitary group $U .$ Let $a = (a_{1}, ..., a_{k})$ and $b = (b_{1}, ..., b_{k})$ two $k -$ tuples of elements in $\A.$ The elementary operator associated to $a$ and $b$ is defined by $R_{a, b} (x) = \sum_{i = 1}^{k} a_{i} x b_{i} .$ In this paper we prove the following formula for the numerical range of $R_{a, b}$ : $$V(R_{a, b}, B(\A))= [\cup \{ V(\sum_{i=1}^ {k}u^*a_iub_i, \A): u\in U\} ]^-.$$ This formulat solves the problem 4.5 of \cite{Fi}.