  The motivation behind this paper is threefold. Firstly, to study,
characterize and realize operator concavity along with its applications to
operator monotonicity of free functions on operator domains that are not
assumed to be matrix convex. Secondly, to use the obtained Schur complement
based representation formulas to analytically extend operator means of
probability measures and to emphasize their study through random variables.
Thirdly, to obtain these results in a decent generality. That is, for domains
in arbitrary tensor product spaces of the form
$\mathcal{A}\otimes\mathcal{B}(E)$, where $\mathcal{A}$ is a Banach space and
$\mathcal{B}(E)$ denotes the bounded linear operators over a Hilbert space $E$.
Our arguments also apply when $\mathcal{A}$ is merely a locally convex space.
