The algebra $A$ is embedded in $A^+$ as $A\times\{0\}$, and the unit of $A^+$ is $(0,1)$.
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If $B$ is a unital $C^*$-algebra and $\pi:A\to B$ is a $*$-homomorphism, then there exists a unique unital $*$-homomorphism $\pi^+:A^+\to B$ such that $\pi^+(a,0)=\pi(a)$ for every $a\in A$. It is given by