The author defines the $\Gamma$ equivariant form of Berezin quantization, where $\Gamma$ is a discrete lattice in $PSL(2, \mathbb R)$. The $\Gamma$ equivariant form of the quantization corresponds to a deformation of the space $\mathbb H/\Gamma$ ($\mathbb H$ being the upper halfplane). The von Neumann algebras in the deformation (obtained via the Gelfand-Naimark-Segal construction from the trace) are type $II_1$ factors. When $\Gamma$ is $PSL(2, \mathbb Z)$, these factors correspond (in the setting considered by K. Dykema and independently by the author, based on the random matrix model of D. Voiculescu) to free group von Neumann algebras with a “fractional number of generators”. The number of generators turns out to be a function of Planck's deformation constant. The Connes cyclic $2$-cohomology associated with the deformation is analyzed and turns out to be (by using an automorphic forms construction) the coboundary of an (unbounded) cycle.

Readership

Graduate students, research mathematicians, and mathematical physicists working in operator algebras.