\(C^*\)-algebras of Toeplitz type associated with algebraic number fields (Q2377366)

To any ring \(R\) of algebraic integers, one associates the \(C^*\)-envelope of left regular representations of the corresponding affine semi-group \(``ax+b"= R\rtimes R^\times\) on the Hilbert space \(\ell^2(R\rtimes R^\times)\). This algebra carries a one-parameter automorphism group \((\sigma_t)_{t\in \mathbb R}\). The KMS-structure is defined. From the authors' summary: ``The technical difficulties that we encounter are due to the presence of the class group in the case where \(R\) is not a principal ideal domain. The ``partition functions'' are partial Dedekind \(\zeta\)-functions. We prove a result characterizing the asymptotic behavior of quotients of such partial \(\zeta\)-functions, which we then use to show uniqueness of the \(\beta\)-KMS state for each inverse temperature \(\beta\in (1,2]\).''