Memoirs of the American Mathematical Society 1998; 70 pp; softcover Volume: 133 ISBN10: 0821807528 ISBN13: 9780821807521 List Price: US$47 Individual Members: US$28.20 Institutional Members: US$37.60 Order Code: MEMO/133/630
 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 GelfandNaimarkSegal 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. Table of Contents  Introduction
 Definitions and outline of the proofs
 Berezin quantization of the upper half plane
 Smooth algebras associated to the Berezin quantization
 The Berezin quantization for quotient space \(\mathbb H/\Gamma\)
 The covariant symbol in invariant Berezin quantization
 A cyclic 2cocycle associated to a deformation quantization
 Bounded cohomology and the cyclic 2cocycle of the Berezin's deformation quantization
 Bibliography
