The Normed Ordered Cone of Operator Connections

A connection in Kubo-Ando sense is a binary operation for positive operators on a Hilbert space satisfying the monotonicity, the transformer inequality and continuity from above. A mean is a connection $σ$ such that $AσA = A$ for all positive operators $A.$ In this paper, we consider the interplay between the cone of connections, the cone of operator monotone functions on the nonnegative reals $\mathbb{R}^+$ and the cone of finite Borel measures on $[0, \infty].$ The set of operator connections is shown to be isometrically order-isomorphic, as normed ordered cones, to the set of operator monotone functions on $\mathbb{R}^+.$ This set is isometrically isomorphic, as normed cones, to the set of finite Borel measures on $[0, ∞].$ It follows that the convergences of the sequence of connections, the sequence of their representing functions and the sequence of their representing measures are equivalent. In addition, we obtain characterizations for a connection to be a mean. In fact, a connection is a mean if and only if it has norm 1 .