Let $H, K$ be Hilbert spaces. Let $A\subseteq B(H)$ be a nonselfadjoint closed subalgebra such that the identity map is in $A$. Let $C_A$ denote the $C^*$-algebra generated by $A$.

Q1: (this question may be obvious to the experts) Let $\pi:A\to B(K)$ be a continuous homomorphism, which is a *-homomorphism restricted to $\{x\in A: x^*\in A\}$. Is $\pi$ completely contractive?

Q2: If $\textrm{socle}(A)=\{0\}$, is it true that $\textrm{socle}(C_A)=\{0\}$?

Q3: Let $C_{env}^*(A)$ denote the $C^*$-envelope of $A$. If $\textrm{socle}(A)=\{0\}$, is it true that $\textrm{socle}(C_{env}^*(A))=\{0\}$?