The parallel sum in C*-algebras

Abstract

Let A be a unital C ∗ -algebra with unit e and let a:b be the parallel sum of the two positive definite elements a and b of A defined by a ( a + b ) − 1 b . We show that the parallel sum a:b can be stated by the solution of the following variational problem: a : b = min x + y = e ( x ∗ ax + y ∗ by ) . This expression leads us to present some properties of the parallel sum in C ∗ -algebras and prove inequalities involving it.