ON GENERALIZED UNIVERSAL IRRATIONAL ROTATION ALGEBRAS AND ASSOCIATED STRONGLY IRREDUCIBLE OPERATORS

Abstract

We introduce a class of generalized universal irrational rotation C*-algebras Aθ, γ = C*(x, w) which is characterized by the relations w*w = ww* = 1, x*x = γ(w), xx* = γ(e-2πiθw), and xw = e-2πiθwx, where θ is an irrational number and γ(z) ∈ C(𝕋) is a positive function. We characterize tracial linear functionals, simplicity, and K-groups of Aθ, γ in terms of zero points of γ(z). We show that if Aθ, γ is simple then Aθ, γ is an A𝕋-algebra of real rank zero. We classify Aθ, γ in terms of θ and zero points of γ(z). Let Aθ = C*(u, v) be the universal irrational rotation C*-algebra with vu = e2πiθuv. Then C*(u + v) ≅ Aθ,|1+z|2. As an application, we show that C*(u + v) is a proper simple C*-subalgebra of Aθ which has a unique trace, K1(C*(u + v)) ≅ ℤ, and there is an order isomorphism of K0(C*(u + v)) onto ℤ + ℤθ. Moreover, C*(u + v) is a unital simple A𝕋-algebra of real rank zero. We also show that u + v is strongly irreducible relative to the hyperfinite type II 1 factor.