  We introduce from an analytic perspective Christoffel-Darboux kernels
associated to bounded, tracial noncommutative distributions. We show that
properly normalized traces, respectively norms, of evaluations of such kernels
on finite dimensional matrices yield classical plurisubharmonic functions as
the degree tends to infinity, and show that they are comparable to certain
noncommutative versions of the Siciak extremal function. We prove estimates for
Siciak functions associated to free products of distributions, and use the
classical theory of plurisubharmonic functions in order to propose a notion of
support for noncommutative distributions. We conclude with some conjectures and
numerical experiments.
