Cohomology of ${\mathcal{A}}_{\theta }^{\mathrm{alg}}⋊{\mathbb{Z}}_2$ and its Chern–Connes pairing

Abstract

We calculate the Hochschild and cyclic cohomology of the noncommutative ${\mathbb{Z}}_2$ toroidal algebraic orbifold ${\mathcal{A}}_{\theta }^{\mathrm{alg}}⋊{\mathbb{Z}}_2$. We also calculate the Chern–Connes pairing of the even periodic cyclic cocycles with the known elements of $K_0({\mathcal{A}}_{\theta }^{\mathrm{alg}}⋊{\mathbb{Z}}_2)$.