Abstract
The injective envelope I(A) of a {C^ * }-algebra A is a unique minimal injective {C^ * }-algebra containing A. As a dynamical system version of the injective envelope of a {C^ * }-algebra we show that for a {C^ * }-dynamical system (A, G, β) with G discrete there is a unique maximal {C^ * }-dynamical system (B, G, β) “containing” (A, G, α) so that A × α rG \subset B × β rG \subset I(A × α rG), where A × α rG is the reduced {C^ * }-crossed product of A by G. As applications we investigate the relationship between the original action α on A and its unique extension I(α) to I(A). In particular, a *-automorphism α of A is quasi-inner in the sense of Kishimoto if and only if I(α) is inner.
