Let Z denote the simple limit of prime dimension drop interval algebras that has a unique tracial state (constructed earlier by Jiang and Su). In this paper, we consider unital C^*-algebras that are isomorphic to their tensor products with Z. We prove that, for such C^*-algebras, the homotopy groups of their unitary groups are given by K-theory. Moreover, such C^*-algebras have cancellation for full projections, and satisfy the comparability question for full projections. Analogous results hold also in the non-unital case.