Hermitian Pairings from Positive Functionals

Question

Let $A$ be $*$-algebra and $\phi:A \to {\mathbb C}$ a positive linear functional, that is, one for which $\phi(aa^*) \geq 0$, for all $a \in A$. When does it hold that a symmetric sesquilinear form, i.e. a Hermitian form, is given by $$ A \times A \to {\mathbb C}, ~~~~~~ (a,b) \mapsto \phi(ab^*). $$ (Note that I am not assuming any completeness here, despite the tags.)

Answer 1

For the start make note: $$ \mathbb{R} \ni \phi((a+1)(a+1)^*) = \phi(aa^*)+\phi(a)+\phi(a^*)+ 1. $$ Next on, since $\phi(aa^*) \in {\mathbb R}$, we must have $\phi(a)+\phi(a^*) \in \mathbb{R}$. Hence, $\Im(\phi(a)) = - \Im(\phi(a^*))$. Do the same trick for $ia$: $$ \mathbb{R} \ni \phi((ia+1)(ia+1)^*) = \phi(aa^*)+i\phi(a) - i\phi(a^*)+ 1 \\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= \phi(aa^*) + 1 + i(\phi(a) - \phi(a^*)). $$ Hence, we have got $$ \Re(\phi(a)) = \Re(\phi(a^*)). $$ We must have at the last $$ \phi(a^*) = \overline{\phi(a)}. $$