Injective envelopes of C∗-dynamical systems

Abstract

The injective envelope 1(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, α) with G discrete there is a unique maximal C∗-dynamical system (B, G,β) “containing” (A, G, α) so that A×arG⊂B×βrG⊂I(A×arG), where A×arG is the reduced C∗-crossed product of A by G. As applications we investigate the relationship between the original action a 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 1(α) is inner. © 1985 Tohoku University, Mathematical Institute.