  We study the Cahen-Gutt moment map on the space of symplectic connections of
a symplectic manifold. On a K\"ahler manifold, we define a Calabi-type
functional $\mathscr{F}$ on the space of K\"ahler metrics in a given K\"ahler
class. We study the zeroes of $\mathscr{F}$. Given a zero of $\mathscr{F}$ with
non-negative Ricci tensor, we show the space of zeroes around the given one has
the structure of a finite dimensional embedded submanifold. We give a new
motivation, coming from deformation quantisation, for the study of moment maps
on infinite dimensional spaces. More precisely, we establish a strong link
between trace densities for star products (obtained from Fedosov's type
methods) and moment map geometry on infinite dimensional spaces. As a
consequence, we provide, on certain K\"ahler manifolds, a geometric
characterization of a space of Fedosov's star products that are closed up to
order 3 in $\nu$.
