Free pluriharmonic functions on noncommutative polyballs

Abstract

We study free $k$-pluriharmonic functions on the noncommutative regular polyball $B_n$, $n=\left(n_1,\dots ,n_k\right)\in {\mathbb{N}}^k$, which is an analogue of the scalar polyball ${\left({ℂ}^{n_1}\right)}_1\times \cdots \times {\left({ℂ}^{n_k}\right)}_1$. The regular polyball has a universal model $S:=\left\{S_{i,j}\right\}$ consisting of left creation operators acting on the tensor product $F^2\left(H_{n_1}\right)\otimes \cdots \otimes F^2\left(H_{n_k}\right)$ of full Fock spaces. We introduce the class ${\mathcal{T}}_n$ of $k$-multi-Toeplitz operators on this tensor product and prove that ${\mathcal{T}}_n=span{\left\{{\mathcal{A}}_n^{\ast }{\mathcal{A}}_n\right\}}^{\text{ -}SOT}$, where ${\mathcal{A}}_n$ is the noncommutative polyball algebra generated by $S$ and the identity. We show that the bounded free $k$-pluriharmonic functions on $B_n$ 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 $B_n^{-}$ if and only if it is the noncommutative Berezin transform of a $k$-multi-Toeplitz operator in $span{\left\{{\mathcal{A}}_n^{\ast }{\mathcal{A}}_n\right\}}^{\text{ -}\parallel \cdot \parallel }$.

We provide a Naimark-type dilation theorem for direct products ${\mathbb{F}}_{n_1}^{+}\times \cdots \times {\mathbb{F}}_{n_k}^{+}$ 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 Berezin (resp. Poisson) transform of a completely bounded linear map on $C^{\ast }\left(S\right)$, the $C^{\ast }$-algebra generated by $S_{i,j}$, and give necessary and 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 the Korányi–Pukánszky version in scalar polydisks.