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Ruelle Operators: Functions which Are Harmonic with Respect to a Transfer Operator
Palle E. T. Jorgensen, University of Iowa, Iowa City, IA
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Memoirs of the American Mathematical Society
2001; 60 pp; softcover
Volume: 152
ISBN-10: 0-8218-2688-3
ISBN-13: 978-0-8218-2688-1
List Price: US$46
Individual Members: US$27.60
Institutional Members: US$36.80
Order Code: MEMO/152/720
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Let \(N\in\mathbb{N}\), \(N\geq2\), be given. Motivated by wavelet analysis, we consider a class of normal representations of the \(C^{\ast}\)-algebra \(\mathfrak{A}_{N}\) on two unitary generators \(U\), \(V\) subject to the relation \(UVU^{-1}=V^{N}\). The representations are in one-to-one correspondence with solutions \(h\in L^{1}\left(\mathbb{T}\right)\), \(h\geq0\), to \(R\left(h\right)=h\) where \(R\) is a certain transfer operator (positivity-preserving) which was studied previously by D. Ruelle. The representations of \(\mathfrak{A}_{N}\) may also be viewed as representations of a certain (discrete) \(N\)-adic \(ax+b\) group which was considered recently by J.-B. Bost and A. Connes.

Readership

Graduate students and research mathematicians interested in functional analysis.

Table of Contents

  • Introduction
  • A discrete \(ax+b\) group
  • Proof of Theorem 2.4
  • Wavelet filters
  • Cocycle equivalence of filter functions
  • The transfer operator of Keane
  • A representation theorem for \(R\)-harmonic functions
  • Signed solutions to \(R(f)=f\)
  • Bibliography
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