Some Formulae for Norms of Elementary Operators

TL;DR

This paper introduces a new formula for calculating the norm of elementary operators on C*-algebras using matrix numerical ranges and a novel tracial geometric mean, also characterizing their compactness.

Contribution

It provides a new formula involving matrix numerical ranges and the tracial geometric mean, and characterizes when elementary operators are compact.

Findings

Derived a new norm formula for elementary operators

Introduced and studied properties of the tracial geometric mean

Characterized the compactness of elementary operators

Abstract

We present a formula for the norm of an elementary operator on a C*-algebra that seems to be new. The formula involves (matrix) numerical ranges and a kind of geometrical mean for positive matrices, the tracial geometric mean, which seems not to have been studied previously and has interesting properties. In addition, we characterise compactness of elementary operators.