# 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

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