Memoirs of the American Mathematical Society 2001; 60 pp; softcover Volume: 152 ISBN10: 0821826883 ISBN13: 9780821826881 List Price: US$49 Individual Members: US$29.40 Institutional Members: US$39.20 Order Code: MEMO/152/720
 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 onetoone 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 (positivitypreserving) 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
