PROJECTIVE COVARIANT REPRESENTATIONS OF LOCALLY $C^*$-DYNAMICAL SYSTEMS

Abstract

As a generalization of covariant completely positive maps, we consider (projective) covariant $\alpha$-completely positive maps between locally $C^*$-algebras. We first study (projective) covariant $J$-representations of locally $C^*$-algebras on Krein modules over locally $C^*$-algebras. Secondly, we construct a covariant KSGNS type representation associated with a covariant $\alpha$-completely positive map on a locally $C^*$-algebra, and then study extensions to a locally $C^*$-crossed product of $\alpha$-completely positive maps on a locally $C^*$-algebra. The results provide (projective) covariant representations of a locally $C^*$-crossed product on a Krein module over a locally $C^*$-algebra.