Holomorphic automorphisms of noncommutative polyballs

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.