# Journal of Operator Theory

Volume 76, Issue 2, Fall 2016  pp. 387-448.

Holomorphic automorphisms of noncommutative polyballs

Authors:  Gelu Popescu
Author institution:Department of Mathematics, The University of Texas at San Antonio, San Antonio, TX 78249, U.S.A.

Summary: In this paper, we study free holomorphic functions on regular polyballs ${\bf B_n}$ and provide analogues of several classical results from complex analysis such as: Abel theorem, Hadamard formula, Cauchy inequality, Schwarz lemma, and maximum principle. These results are used together with a class of noncommutative Berezin transforms to obtain a complete description of the group $\text{\rm Aut}({\bf B_n})$ of all free holomorphic automorphisms of the polyball ${\bf B_n}$. We also obtain a concrete description for the group of automorphisms of the tensor product $\mathcal T_{n_1}\otimes\cdots \otimes\mathcal T_{n_k}$ of Cuntz--Toeplitz algebras which leave invariant the tensor product $\mathcal A_{n_1}\otimes_\mathrm{min}\cdots \otimes_\mathrm{min}\mathcal A_{n_k}$ of noncommutative disc algebras, which extends Voiculescu's result when $k=1$.

DOI: http://dx.doi.org/10.7900/jot.2015dec12.2088
Keywords: noncommutative polyball, automorphism, Berezin transform, Fock space, creation operators, Cuntz-Toeplitz algebra