On strong affine representations of the polycyclic monoids

Abstract

Jones and Lawson have discovered in (Period Math Hungar 64(1):53–87 2012) that certain representations of the so-called polycyclic monoids are closely related to some permutative representations of the Cuntz algebras $${\mathcal {O}}_{n}$$ studied by Bratteli and Jorgensen in (Mem Am Math Soc 139(663):x+89 1999). We investigate these representations of the polycyclic monoids, and we generalize some results from Jones and Lawson (2012). We give a (sharp) upper bound on the number of atoms in case one of the parameters tends to infinity and present an infinite family of representations having only one atom. Furthermore, by making use of a C++ program we present some observations regarding the number of atoms in the case $$n=3$$ .