Phase transition in the Connes-Marcolli \(\mathrm{GL}_2\)-system (Q2472578)

Summary: We develop a general framework for analyzing KMS-states on \(C^*\)-algebras arising from actions of Hecke pairs. We then specialize to the system recently introduced by Connes and Marcolli and classify its KMS-states for inverse temperatures \(\beta\neq 0,1\). In particular, we show that for each \(\beta\in(1,2]\) there exists a unique KMS\(_\beta\)-state.