Two parameter $\sigma$-$C^*$-Dynamical systems and application

Abstract

Let $\sigma$ be a bounded linear $*$-endomorphism on the $C^*$-algebra $\mathcal{A}$.Introducing the notions of
$*$-$\sigma$-derivations and two parameter $\sigma$-$C^*$-dynamical
sytems, we correspond to each so-called two parameter
$\sigma$-$C^*$-dynamical system a pair of $\sigma$-derivations,
named as its infinitesimal generator. Also, using the computation
formula for $\sigma$-derivations, we deal with the converse under
some restrictions. Finally, as an interesting application, we
characterize each so-called two parameter $\sigma$-$C^*$-dynamical
system on the concrete $C^*$-algebra $A:=B(\mathcal{H})\times
B(\mathcal{H}),$ where $\mathcal{H}$ is a Hilbert space and $\sigma$
is the linear $*$-endomorphism $\sigma(S,T)=(0,T).$