Definition of $*$-endomorphisms on a $C^*$-algebra

Question

I'm reading a paper which talks about $*$-endomorphisms on a $C^*$-algebra without defining the notion. I browsed through various books on $C^*$-algebras but could not find a definition. I only found a definition of an endomorphism on a measure space $(\Omega,\mu)$.

What is the standard definition of a $*$-endomorphism on a $C^*$-algebra?

Answer 1

The word endomorphism is used to denote a morphism where both the domain and the codomain coincide, meaning a morphism $\varphi : A\to B$ is an endomorphism iff $A=B$. According to wiktionary the root of the "endo-" is the greek $\mathrm{\epsilon\nu\delta\omicron\nu}$, meaning "inner".

For C* algebras the appropriate notion of morphism is that of a $*$-morphism, so here an endomorphism of a C*algebra $A$ is a map $\varphi:A\to A$ satisfying

1. Linearity: $\varphi(\lambda a+ b) = \lambda\varphi(a)+\varphi(b)$ for all $a,b\in A$, $\lambda\in \Bbb K$.
2. Multiplicativity $\varphi(ab)= \varphi(a)\varphi(b)$ for all $a,b\in A$.
3. Compatibility with $*$: $\varphi(a^*)=\varphi(a)^*$ for all $a\in A$.