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Baumslag-Solitar group \(C^*\)-algebras from interval maps. (English) Zbl 1296.46056
Summary: 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.
MSC:
46L55 Noncommutative dynamical systems
46L05 General theory of \(C^*\)-algebras
37B10 Symbolic dynamics
37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations
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