Geometric aspects of similarity problems

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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· 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.