Geometric aspects of similarity problems

TL;DR

This paper explores geometric methods to analyze similarity problems in non-positively curved spaces of positive invertible operators, extending prior results and addressing questions about minimality of canonical unitarizers.

Contribution

It introduces a geometric framework for similarity problems in operator algebras, extending previous results and partially answering open questions.

Findings

Extended and unified geometric approach to similarity problems.

Provided partial answers to minimality properties of canonical unitarizers.

Connected geometric properties with algebraic similarity concepts.

Abstract

This article presents a geometric approach to some similarity problems involving metric arguments in the non-positively curved space of positive invertible operators of an operator algebra and the canonical isometric action by invertible elements on the cone given by $g\cdot a=gag^*$. Through this approach, we extend and put into a geometric framework results by G. Pisier and partially answer a question by Andruchow, Corach and Stojanoff about minimality properties of canonical unitarizers.