On positivity and roots in operator algebras

TL;DR

This paper explores a new notion of positivity in operator algebras, extending C*-algebra results, and investigates properties like the numerical range of roots of commuting operators within this framework.

Contribution

It provides supplementary results on the introduced positivity concept, including a theorem on the numerical range of roots of commuting operators.

Findings

Numerical range of roots of commuting operators in a sector

Extensions of positivity concepts to broader operator algebras

Complementary results to previous positivity studies

Abstract

In earlier papers the second author and Charles Read have introduced and studied a new notion of positivity for operator algebras, with an eye to extending certain C*-algebraic results and theories to more general algebras. The present paper consists of complements to some facts in the just mentioned papers, concerning this notion of positivity. For example we prove a result on the numerical range of products of the roots of commuting operators with numerical range in a sector.