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

Abstract

Let $\sigma$ be a linear $*$-endomorphism on a $C^*$-algebra $A$ so that $\sigma(A)$ acts on a Hilbert space $H$ which including $\mathcal{K}(H)$
and let $\{\alpha_t\}_{t\in\mathbb{R}}$ be a $\sigma$-$C^*$-dynamical system
on $A$ with the generator $\delta.$  In this paper, we
demonstrate some conditions under which $\{\alpha_t\}_{t\in\mathbb{R}}$  is implemented by a
$C_0$-groups of unitaries on $H$.