Let 𝒜 denote the C*-algebra of bounded operators on L2(ℝ × 𝕊1) generated by: all multiplications a(M) by functions a ⋲ C∞(𝕊1), all multiplications b(M) by functions b ⋲ C([−∞,+∞]), all multiplications by 2π-periodic continuous functions, Λ = (1 − Δℝ×𝕊1 )−1/2, where Δℝ×𝕊1 is the Laplacian operator on L2(ℝ × 𝕊1), and ϑtΛ, ϑxΛ, for t ⋲ ℝ and x⋲ 𝕊1. We compute the K-theory of 𝒜 and of its quotient by the ideal of compact operators.