Baumslag-Solitar group C*-algebras from interval maps

Abstract

We yield operators $U$ and $V$ on Hilbert spaces that are parameterized by the orbits of certain interval maps that exhibit chaotic behavior and obey the (deformed) Baumslag--Solitar relation $$UV=e^{2\pi i \alpha} VU^n,\qquad \alpha\in \mathbb{R},\ n\in\mathbb{N}.$$ We then prove that the scalar $e^{2\pi i \alpha}$ can be removed whilst retaining the isomorphism class of the $C^*$-algebra generated by $U$ and $V$. Finally, we simultaneously unitarize $U$ and $V$ by gluing pairs of orbits of the underlying noninvertible dynamical system and investigate these unitary representations under distinct pairs of orbits.