Simple c*-algebras arising from β-expansion of real numbers

Given a real number β > 1, we construct a simple purely infinite C*-algebra script O sign_β as a C*-algebra arising from the β-subshift in the symbolic dynamics. The C*-algebras {script O sign_β}_1<β∈ℝ interpolate between the Cuntz algebras {script O sign_n}_1<n∈ℕ. The K-groups for the C*-algebras script O sign_β, 1 < β ∈ ℝ, are computed so that they are completely classified up to isomorphism. We prove that the KMS-state for the gauge action on script O sign_β is unique at the inverse temperature log β. which is the topological entropy for the β-shift. Moreover, script O sign_β is realized to be a universal C*-algebra generated by n - 1 = [β] isometries and one partial isometry with mutually orthogonal ranges and a certain relation coming from the sequence of β-expansion of 1.