The Christoffel-Darboux kernel is the reproducing kernel associated to the Hilbert space containing all polynomials up to a given degree. It can be naturally written in terms of any complete set of orthonormal polynomials. In classical analysis the Christoffel-Darboux kernel is useful for studying properties of the underlying measure with respect to which the Hilbert space of polynomials is defined. In this talk, we present the version of the Christoffel-Darboux kernel for $L^2$ spaces of tracial states on noncommutative polynomials. We view this kernel as a noncommutative function, and identify its values as maxima of certain sets of  non-negative matrices/operators.

In numerous cases, the classical version of the Christoffel-Darboux kernel can be used (after renormalization) to recover the measure to which it is associated as a weak derivative. This is done with the aid of the theory of plurisubharmonic functions. We use this same theory in order to introduce several noncommutative versions of the Siciak extremal function. We use the Siciak functions to prove that, in several cases of interest, the (properly normalized) limit of the evaluations of the Christoffel-Darboux kernel on matrix sets exists as a well-defined, quasi-everywhere finite plurisubharmonic function. Time permitting, we conclude with some conjectures regarding these objects. This is based on joint work with Victor Magron (LAAS) and Victor Vinnikov (Ben Gurion
University).