FAITHFUL REPRESENTATIONS OF CROSSED-PRODUCTS BY ENDOMORPHISMS

Stacey has recently characterised the crossed product A x(alpha) N of a C*-algebra A by an endomorphism alpha as a C*-algebra whose representations are given by covariant representations of the system (A, alpha). Following work of O'Donovan for automorphisms, we give conditions on a covariant representation (pi, S) of (A, alpha) which ensure that the corresponding representation pi x S of A x(alpha) N is faithful. We then use this result to improve a theorem of Paschke on the simplicity of A x(alpha) N.