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