Generalized Fixed Point Algebras and Square-integrable Group Actions

We analzye Rieffel’s construction of generalized fixed point algebras in the setting of group actions on Hilbert modules. Let G be a locally compact group acting on a C∗-algebra B. We construct a Hilbert module F over the reduced crossed product of G and B, using a pair (E, R), where E is an equivariant Hilbert module over B and R is a dense subspace of E with certain properties. The generalized fixed point algebra is the C∗-algebra of compact operators on F . Any Hilbert module over the reduced crossed product arises by this construction for a pair (E, R) that is unique up to isomorphism. A necessary condition for the existence of R is that E be square-integrable. The consideration of square-integrable representations of Abelian groups on Hilbert space shows that this condition is not sufficient and that different choices for R may yield different generalized fixed point algebras. If B is proper in Kasparov’s sense, there is a unique R with the required properties. Thus the generalized fixed point algebra only depends on E.