On the standard K-loop structure of positive invertible elements in a C⁎-algebra

We investigate the algebraic properties of the operation a∘b=aba on the set of all positive invertible elements of a C⁎-algebra A. We show that its commutativity, associativity and distributivity are each equivalent to the commutativity of A. We present abstract characterizations of the operation ∘ and a few related ones, too.