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

TL;DR

This paper derives a precise formula for the numerical range of elementary operators on C*-algebras, connecting it to unitary conjugations and solving a previously open problem.

Contribution

It provides a new explicit formula for the numerical range of elementary operators on C*-algebras, addressing an open problem in the field.

Findings

Derived a formula for the numerical range of elementary operators

Connected the numerical range to unitary conjugations within the algebra

Solved an open problem from prior literature

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_ixb_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}.