C*-algebraic deformations of homogeneous spaces GΓ1 are constructed by completing dense subspaces of C 0 (GΓ1 ) in a different multiplication and C*-norm; these deformations are equivariant in the sense that they still carry a natural action of G by left translation. The motivating examples are the noncommutative Heisenberg manifolds of Rieffel, but the construction given here is based on the authors' twisted dual-group algebras, which are equivariant deformations of C 0 (G ). The procedure is general enough to give other recently studied families of noncommutative spaces and some interesting new examples of simple C*-algebras.
1997 Academic Press
A C*-algebraic deformation of a locally compact space X (or the algebra C 0 (X)) is a C*-algebra obtained by completing a *-algebra whose underlying vector space is a dense subspace of C 0 (X). It is important in applications that extra structure possessed by X can be carried over to the deformation: thus, for example, one might expect deformations of groups to have comultiplications that is, to be quantum groups. Here we consider deformations of homogeneous spaces GΓ1, and we look for ones which preserve the natural left action of G. Thus we want deformations which are equivariant in the sense that the underlying vector space V is invariant under the action of G by left translation on C 0 (GΓ1 ) and the resulting action of G on V extends to the C*-completion. Our algebras are based on the equivariant deformations of C 0 (G) constructed in our previous paper [LR], and the motivating examples are the Heisenberg manifolds, where our construction yields the deformations obtained by Rieffel [R1].
The deformations of C 0 (G) in [LR] have as underlying vector space the Fourier algebra A(G), and the multiplication is the usual one twisted by article no.