Proposition: Positivity on a common physical algebra

Let $\mathfrak{A}$ be one common unital $\ast$-algebra. If normalized positive functionals ${\omega }_{\Lambda }$ converge pointwise on $\mathfrak{A}$ to ${\omega }_{\infty }$, then ${\omega }_{\infty }$ is linear, normalized, and positive.

The common physical algebra is essential. This is not a positivity theorem for the full gauge-fixed ghost algebra. Hadamard product data alone do not supply the positive functionals in the hypothesis.

Proof. Linearity passes through pointwise limits. Also ${\omega }_{\infty }(1)=1$, and for every $a\in \mathfrak{A}$,