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 continuous if and only if $ f$ is positive 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.).
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