  We develop a theory of abstract intermediate function spaces on a compact
convex set $X$ and study the behaviour of multipliers and centers of these
spaces. In particular, we provide some criteria for coincidence of the center
with the space of multipliers and a general theorem on boundary integral
representation of multipliers. We apply the general theory in several concrete
cases, among others to strongly affine Baire functions, to the space $A_f(X)$
of fragmented affine functions, to the space $(A_f(X))^\mu$, the monotone
sequential closure of $A_f(X)$, to their natural subspaces formed by Borel
functions, or, in some special cases, to the space of all strongly affine
functions. In addition, we prove that the space $(A_f(X))^\mu$ is determined by
extreme points and provide a large number of illustrating examples and
counterexamples.
