  An axiomatic theory of operator connections and operator means was
investigated by Kubo and Ando in 1980. A connection is a binary operation for
positive operators satisfying the monotonicity, the transformer inequality and
the joint-continuity from above. In this paper, we show that the
joint-continuity assumption can be relaxed to some conditions which are weaker
than the separate-continuity. This provides an easier way for checking whether
a given binary opertion is a connection. Various axiomatic characterizations of
connections are obtained. We show that the concavity is an important property
of a connection by showing that the monotonicity can be replaced by the
concavity or the 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
any named means.
