Operator connections and Borel measures on the unit interval

Abstract

A connection is a binary operation for positive operators satisfying monotonicity, the transformer inequality, and joint-continuity from above.A normalized connection is called a mean.Here we show that there is a one-to-one correspondence between connections and finite Borel measures on the unit interval via the integral representation in terms of weighted harmonic means with respect to that measure.This correspondence is affine and order-preserving.Hence every mean can be regarded as an average of weighted harmonic means.We also investigate decompositions of connections, means, symmetric connections and symmetric means.