Non-commutative Markov Processes in Free Groups Factors, Related to Berezin’s Quantization and Automorphic Forms

In this paper we use the description of free group factors as the von Neumann algebras of Berezin’s deformation of the upperhalf plane, modulo PSL2(Z). The derivative, in the deformation parameter, of the product in the corresponding algebras, is a positive — 2 Hochschild cocycle, defined on a dense subalgebra. By analyzing the structure of the cocycle we prove that there is a generator, L, for a quantum dynamical semigroup, that implements the cocycle on a strongly dense subalgebra. For x in the dense subalgebra, L(x) is the (diffusion) operator L(x) = Λ(x)− 1/2{T, x}, where Λ is the pointwise (Schurr) multiplication operator with a symbol function related to the logarithm of the automorphic form ∆. T corresponds to L(1), in a sense to be made precise in the paper. After a suitable normalization, corresponding to a principal value type method, adapted for II1 factors, Λ becomes (completely) positive. Moreover the 2cyclic cohomology cocycle associated to the deformation may be expressed in terms of Λ