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The \(\Gamma\)-Equivariant Form of the Berezin Quantization of the Upper Half Plane
Florin Rădulescu, University of Iowa, Iowa City, IA
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Memoirs of the American Mathematical Society
1998; 70 pp; softcover
Volume: 133
ISBN-10: 0-8218-0752-8
ISBN-13: 978-0-8218-0752-1
List Price: US$44
Individual Members: US$26.40
Institutional Members: US$35.20
Order Code: MEMO/133/630
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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 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

  • 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 2-cocycle associated to a deformation quantization
  • Bounded cohomology and the cyclic 2-cocycle of the Berezin's deformation quantization
  • Bibliography
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