Mathematics – Operator Algebras
Scientific paper
1999-12-04
Mathematics
Operator Algebras
85 pages, Plain TeX. Changes: typographical errors have been fixed; also, extensions in Sections 5 and 6
Scientific paper
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 $\Lambda$.
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