Item Successfully Added to Cart
An error was encountered while trying to add the item to the cart. Please try again.
OK
Please make all selections above before adding to cart
OK
Share this page via the icons above, or by copying the link below:
Copy To Clipboard
Successfully Copied!
The $\Gamma$-Equivariant Form of the Berezin Quantization of the Upper Half Plane
 
Florin Ră dulescu University of Iowa, Iowa City, IA
The Gamma-Equivariant Form of the Berezin Quantization of the Upper Half Plane
eBook ISBN:  978-1-4704-0219-8
Product Code:  MEMO/133/630.E
List Price: $47.00
MAA Member Price: $42.30
AMS Member Price: $28.20
The Gamma-Equivariant Form of the Berezin Quantization of the Upper Half Plane
Click above image for expanded view
The $\Gamma$-Equivariant Form of the Berezin Quantization of the Upper Half Plane
Florin Ră dulescu University of Iowa, Iowa City, IA
eBook ISBN:  978-1-4704-0219-8
Product Code:  MEMO/133/630.E
List Price: $47.00
MAA Member Price: $42.30
AMS Member Price: $28.20
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1331998; 70 pp
    MSC: 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 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.

  • 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 2-cocycle associated to a deformation quantization
    • 6. Bounded cohomology and the cyclic 2-cocycle of the Berezin’s deformation quantization
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1331998; 70 pp
MSC: 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 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.

  • 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 2-cocycle associated to a deformation quantization
  • 6. Bounded cohomology and the cyclic 2-cocycle of the Berezin’s deformation quantization
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.