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.).

  • $\begingroup$ It seems that there is no general theory even for fairly decent classes of topological algebras as suggest recent proofs for particular cases; see, for example, sciencedirect.com/science/article/pii/S0022247X15006307 $\endgroup$ – Tomek Kania Dec 15 '17 at 15:55
  • $\begingroup$ The question makes no sense. $f$ is continuous implies $f$ is positive? $\endgroup$ – Nik Weaver Dec 15 '17 at 17:48
  • $\begingroup$ @NikWeaver: On unital $\mathbb C$-$*$-algebras yes, with the condition $\| f \| = f(1)$ that you may have involuntarily skipped. See the bibliography referenced on the relevant Wikipedia page. Intuitively, non-rigorously, $f$ is an "integral", therefore it is associated to a positive measure, therefore it must be positive. How do we deal with the downvote now? $\endgroup$ – Alex M. Dec 15 '17 at 17:54
  • $\begingroup$ If $f$ is linear and continuous then so is $-f$. $\endgroup$ – Nik Weaver Dec 15 '17 at 17:58
  • $\begingroup$ @NikWeaver: Mind slip, obviously: I had interchanged the words "positive" and "continuous". Why the need to downvote such minor mistakes? $\endgroup$ – Alex M. Dec 15 '17 at 18:03

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