Geometric aspects of similarity problems

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.