Let $ N \in \mathbb{N} $, $ N \geq 2 $, be given. Motivated by wavelet
analysis, we consider a class of normal representations of the $ C^* $-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}(\mathbb{T}) $, $ h \geq 0 $, to $
R(h)=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.
@article{9805141,
author = {Jorgensen, Palle E. T.},
title = {Ruelle operators: Functions which are harmonic with respect to a
transfer operator},
journal = {arXiv},
volume = {1998},
number = {0},
year = {1998},
language = {en},
url = {http://dml.mathdoc.fr/item/9805141}
}
Jorgensen, Palle E. T. Ruelle operators: Functions which are harmonic with respect to a
transfer operator. arXiv, Tome 1998 (1998) no. 0, . http://gdmltest.u-ga.fr/item/9805141/