2015/02/27 by Popescu, Gelu
#FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.1502.07905
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(Bn) of all free holomorphic automorphisms of the polyball. The abstract polyball Bn 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 An; the group of unitarily implemented automorphisms of the polyball algebra An and the noncommutative Hardy algebra Fn^∞, respectively. We prove that the free holomorphic automorphism group Aut(Bn) 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(Bn)in terms of noncommutative Berezin kernels associated with regular polyballs.