eBook ISBN:  9781470402198 
Product Code:  MEMO/133/630.E 
List Price:  $47.00 
MAA Member Price:  $42.30 
AMS Member Price:  $28.20 
eBook ISBN:  9781470402198 
Product Code:  MEMO/133/630.E 
List Price:  $47.00 
MAA Member Price:  $42.30 
AMS Member Price:  $28.20 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 133; 1998; 70 ppMSC: Primary 46; Secondary 81; 11
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.
ReadershipGraduate students, research mathematicians, and mathematical physicists working in operator algebras.

Table of Contents

Chapters

Introduction

0. Definitions and outline of the proofs

1. Berezin quantization of the upper half plane

2. Smooth algebras associated to the Berezin quantization

3. The Berezin quantization for quotient space $\mathbb {H}/\Gamma $

4. The covariant symbol in invariant Berezin quantization

5. A cyclic 2cocycle associated to a deformation quantization

6. Bounded cohomology and the cyclic 2cocycle of the Berezin’s deformation quantization


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

Chapters

Introduction

0. Definitions and outline of the proofs

1. Berezin quantization of the upper half plane

2. Smooth algebras associated to the Berezin quantization

3. The Berezin quantization for quotient space $\mathbb {H}/\Gamma $

4. The covariant symbol in invariant Berezin quantization

5. A cyclic 2cocycle associated to a deformation quantization

6. Bounded cohomology and the cyclic 2cocycle of the Berezin’s deformation quantization