Non-Commutative Markov Processes in Free Group 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 upper half-plane, modulo PSL$(2,{\Bbb Z})$. The derivative, in the deformation parameter, of the product in the corresponding algebras, is a positive Hochschild 2-cocycle, defined on a dense subalgebra. By analyzing the structure of the cocycle we prove that there is a generator $\cal L$ for a quantum dynamical semigroup that implements the cocycle on a strongly dense subalgebra. For $x$ in the dense subalgebra, ${\cal L}(x)$ is the (diffusion) operator $${\cal L}(x)=\Lambda(x)-(1/2)\{T,x\},$$ where $\Lambda$ is the pointwise (Schur) multiplication operator with a symbol function related to the logarithm of the automorphic form $\Delta$. The operator $T$ is positive and affiliated with the algebra ${\cal A}_t$ and $T$ corresponds to ${\cal L}(1)$, in a sense to be made precise in the paper. After a suitable normalization, corresponding to a principal-value type method, adapted for II$_1$ factors, $\Lambda$ becomes (completely) positive on a union of weakly dense subalgebras. Moreover the 2-cyclic cohomology cocycle associated to the deformation may be expressed in terms of $\Lambda$.