Characterizations of connections for positive operators.

*(English)*Zbl 1299.47076Summary: In this paper, we investigate the relationships among algebraic, order and topological properties of a binary operation for positive operators. A connection is a binary operation satisfying monotonicity, the transformer inequality and joint-continuity from above. We show that the joint-continuity assumption can be relaxed to some conditions that are weaker than separate-continuity. Various axiomatic characterizations of connections are obtained. We show that concavity is an important property of a connection by showing that monotonicity can be replaced by concavity or midpoint concavity. Each operator connection induces a unique scalar connection. Moreover, there is an affine order isomorphism between connections and induced connections. This gives a natural viewpoint to define means.