Operator Connections and Borel Measures on the Unit Interval
  A connection is a binary operation for positive operators satisfying the monotonicity, the transformer inequality and the joint-continuity from above. A mean is a normalized connection. In this paper, we show that there is a one-to-one correspondence between connections and finite Borel measures on the unit interval via a suitable integral representation. Every mean can be regarded as an average of weighted harmonic means. Moreover, we investigate decompositions of connections, means, symmetric connections and symmetric means. 
Pattrawut Chansangiam, Wicharn Lewkeeratiyutkul
math.FA