Free Pluriharmonic Functions on Noncommutative Polyballs

Abstract: In this paper, we study free k-pluriharmonic functions on noncommutative regular polyballs. These regular polyballs have universal operator models consisting of left creation operators acting on tensor products of full Fock spaces. We introduce and determine the class of kmulti- Toeplitz operators acting on these tensor products and show that the bounded free k-pluriharmonic functions on regular polyballs are precisely the noncommutative Berezin transforms of k-multi-Toeplitz operators. The Dirichlet extension problem on regular polyballs is also solved. It is proved that a free k-pluriharmonic function has continuous extension to the closed polyball if and only if it is the noncommutative Berezin transform of a k-multi-Toeplitz operator in a certain class, which we determine. We provide a Naimark type dilation theorem for direct products of unital free semigroups, and use it to obtain a structure theorem which characterizes the positive free k-pluriharmonic functions on the regular polyball, with operator-valued coefficients. We define the noncommutative the Berezin (resp. Poisson) transform of a completely bounded linear map on the C*-algebra generated by the universal operator model and give necessary an sufficient conditions for a function to be the Poisson transform of a completely bounded (resp. completely positive) map. In the last section of the paper, we obtain Herglotz-Riesz representation theorems for free holomorphic functions on regular polyballs with positive real parts, extending the classical result as well as Korannyi-Pukanszky version in scalar polydisks.