2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/79544In 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 $$ {\cal L}(x)=Λ(x)-(1/2)\{T,x\}, $$ where $Λ$ is the pointwise (Schur) multiplication operator with a symbol function related to the logarithm of the automorphic form $Δ$. 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, $Λ$ 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 $Λ$.85 pages, Plain TeX. Changes: typographical errors have been fixed; also, extensions in Sections 5 and 6Operator AlgebrasQuantum Algebratext