Let $A$ be a normed $*$-algebra, let $\alpha$ be a continuous positive form on $A$, which has finite variation, meaning there exists $C>0$ such that $|\alpha (a)|^2 \leq C \cdot \alpha (a^* a)$ for all $a \in A$. Do then necessarily exist a Hilbert space $H$, a continuous morphism of $*$-algebras $\pi : A \to \mathcal{B} (H)$, and a vector $v \in H$ such that $\alpha (v) = \langle av , v \rangle$ for all $a \in A$?
I think one can pass to the completion, thus assuming that $A$ is a Banach $*$-algebra. Then it is not necessary to ask for $\pi$ to be continuous, as it will automatically be so.
If $A$ has approximate identity (that is, a net $(u_i)_{i \in I}$ such that $|| u_i || \leq 1$, $|| u_i a - a || \to 0$, $|| a u_i - a || \to 0$), then classically the answer is positive.
Thanks