σ-c∗-dynamics of k(h)

Let σ be a linear ∗-endomorphism on a c ∗ -algebra a so that σ(a) acts on a hilbert space h which including k(h) and let {αt}t∈r be a σ-c ∗ -dynamical system on a with the generator δ. in this paper, we demonstrate some conditions under which {αt}t∈r is implemented by a c0-groups of unitaries on h. in particular, we prove that for a rankone projection p ∈ a, which is invariant under αt, there is a c0-group {ut}t∈r of unitaries in b(h) such that αt(a) = utσ(a)u ∗ t . furthermore, introducing the concepts of σ-inner endomorphism and σ-bijective map, we prove that each σ-bijective linear endomorphism on a is a σ-inner endomorphism, where σ ia idempotent. finally, as an application, we characterize each so-called σ-c ∗ -dynamical system on the concrete c ∗ - algebra a := b(h) × b(h), where h is a separable hilbert space and σ is the linear ∗-endomorphism σ(s, t) = (0, t) on a.