In this paper we study the unitary equivalence between Hilbert modules over a locally C * -algebra. Also, we prove a stabilization theorem for countably generated modules over an arbitrary locally C * -algebra and show that a Hilbert module over a Fréchet locally C * -algebra is countably generated if and only if the locally C * -algebra of all "compact" operators has an approximate unit.2000 Mathematics Subject Classification: 46L08, 46L05

The structure of the continuous strict completely positive linear maps between locally C * -algebras is described.

We prove a covariant version of the Stinespring theorem for Hilbert C * -modules.2000 Mathematics Subject Classification. Primary 46L08.

The crossed products of locally C * -algebras are defined and a Takai duality theorem for inverse limit actions of a locally compact group on a locally C * -algebra is proved.2000 AMS Mathematics subject classification. Primary 46L05, 46L55.

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