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2016 Free pluriharmonic functions on noncommutative polyballs
Gelu Popescu
Anal. PDE 9(5): 1185-1234 (2016). DOI: 10.2140/apde.2016.9.1185


We study free k-pluriharmonic functions on the noncommutative regular polyball Bn, n = (n1,,nk) k , which is an analogue of the scalar polyball (n1)1 × × (nk)1. The regular polyball has a universal model S := {Si,j} consisting of left creation operators acting on the tensor product F2(Hn1) F2(Hnk) of full Fock spaces. We introduce the class Tn of k-multi-Toeplitz operators on this tensor product and prove that T n = span{AnAn} - SOT, where An is the noncommutative polyball algebra generated by S and the identity. We show that the bounded free k-pluriharmonic functions on Bn 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 Bn if and only if it is the noncommutative Berezin transform of a k-multi-Toeplitz operator in span{AnAn} -.

We provide a Naimark-type dilation theorem for direct products Fn1+ × × Fnk+ 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(S), the C-algebra generated by Si,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.


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Gelu Popescu. "Free pluriharmonic functions on noncommutative polyballs." Anal. PDE 9 (5) 1185 - 1234, 2016.


Received: 3 December 2015; Revised: 29 February 2016; Accepted: 12 April 2016; Published: 2016
First available in Project Euclid: 12 December 2017

zbMATH: 1353.47007
MathSciNet: MR3531370
Digital Object Identifier: 10.2140/apde.2016.9.1185

Primary: 47A13 , 47A56
Secondary: 46L52 , 47B35

Keywords: ‎Berezin transform , completely bounded map , Fock space , free holomorphic function , Herglotz–Riesz representation , multi-Toeplitz operator , Naimark dilation , noncommutative polyball , pluriharmonic function , Poisson transform

Rights: Copyright © 2016 Mathematical Sciences Publishers


Vol.9 • No. 5 • 2016
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