$\sigma$-$C^*$-dynamics of $\mathcal{K}(H)$

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 rank- one 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.