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

We investigate the algebraic properties of the operation a ∘ b = a b a 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.