Irrational rotation algebra: Let the infinite cyclic group Z act on the circle S1 by the rotation action by angle 2iθ.
Twisted group algebra: The function σ : Z2 × Z2 → C; σ((m,n), (p,q)) = e2πinpθ is a group 2-cocycle on Z2, and the corresponding twisted group algebra C*(Z2; σ) is isomorphic to Aθ.