It is known that if $\mathcal A$ is a unital $\mathbb C$-$*$-algebra and $A$ is a unital subalgebra closed under $*$, and if $f : A \to \mathbb C$ is linear, then $f$ is positive if and only if $f$ is continuous and $\|f\| = f(1)$.

Can similar statements be produced for a larger class of topological algebras? I am particularly interested in the case when $\mathcal A$ is the algebra $C_b (X)$ of bounded continuous functions on some Hausdorff topological space $X$, endowed with some of the the usual interesting topologies given by modes of convergence (compact convergence, strict topology etc.).