# The \(\Gamma\)-Equivariant Form of the Berezin Quantization of the Upper Half Plane

Share this page
*Florin Ră dulescu*

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.

#### Table of Contents

# Table of Contents

## The $$-Equivariant Form of the Berezin Quantization of the Upper Half Plane

- Contents vii8 free
- Introduction 110 free
- Chapter 0. Definitions and outline of the proofs 312 free
- Chapter 1. Berezin quantization of the upper half plane 1019
- Chapter 2. Smooth algebras associated to the Berezin quantization 1625
- Chapter 3. The Berezin quantization for quotient space H/T 2534
- Chapter 4. The covariant symbol in invariant Berezin quantization 3847
- Chapter 5. A cyclic 2-cocycle associated to a deformation quantization 5059
- Chapter 6. Bounded cohomology and the cyclic 2-cocycle of the Berezin's deformation quantization 5867
- Bibliography 6978