In this paper, we study free holomorphic functions on regular polyballs 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 Aut(B_n) of all free holomorphic automorphisms of the polyball. The abstract polyball B_n has a universal model S consisting of left creation operators acting on tensor products of full Fock spaces. We determine: the group of automorphisms of the Cuntz-Toeplitz algebra C*(S) which leaves invariant the noncommutative polyball algebra A_n; the group of unitarily implemented automorphisms of the polyball algebra A_n and the noncommutative Hardy algebra F_n^\infty, respectively. We prove that the free holomorphic automorphism group Aut(B_n) is a sigma-compact, locally compact topological group with respect to the topology induced by an appropriate metric. Finally, we obtain a concrete unitary projective representation of the topological group Aut(B_n)in terms of noncommutative Berezin kernels associated with regular polyballs.
本文得到了如下算子值多元Bohr型不等式：一类非交换全纯函数，推广了开单位圆盘上的解析函数;非交换圆盘代数和非交换解析Toeplitz代数;一类非交换自伴调和函数，推广了开单位圆盘上的实值调和函数;具有n个生成元的自由群的约化（或全）群C* 代数; C^n的开单位球上的一类解析函数。证明了经典的Bohr不等式是正三角多项式系数的Fejer不等式和幂零算子幂零算子的Haagerup-de la Harpe不等式的一个推论.此外，我们提供了一个不等式，它是开放的单位圆盘上的解析多项式，比玻尔的不等式更尖锐。
