  A connection in Kubo-Ando sense is a binary operation for positive operators
on a Hilbert space satisfying the monotonicity, the transformer inequality and
the continuity from above. A mean is a connection $\sigma$ such that $A \sigma
A =A$ for all positive operators $A$. In this paper, we consider the interplay
between the cone of connections, the cone of operator monotone functions on
$\R^+$ and the cone of finite Borel measures on $[0,\infty]$. %We define a norm
for a connection in such a way that the set of operator connections becomes %a
normed ordered cone. %On the other hand, the cone of operator monotone
functions on $\R^+$ %and the cone of finite Borel measures on $[0,\infty]$ are
equipped with suitable norms. The set of operator connections is shown to be
isometrically order-isomorphic, as normed ordered cones, to the set of operator
monotone functions on $\R^+$. This set is isometrically isomorphic, as normed
cones, to the set of finite Borel measures on $[0,\infty]$. 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.

Functional Analysis