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I am trying to understand what (minimal) conditions one would need in order to obtain a functional calculus on a Fréchet algebra, which we demand to be equipped with an involution that leaves all semi-norms invariant.

The motivation behind this is that I need to see if forcing a Fréchet algebra to have positive elements similar to a ${C^{*}}$-algebra leaves me with sufficiently interesting Fréchet algebras. Having a continuous calculus would, I hope, be enough to give me the kind of $C^{*}$-behaviour I require on positive elements.

I understand that a holomorphic functional calculus can be developed if the algebra is submultiplicative in each semi-norm, but to follow the classical route for a continuous calculus along $C^{*}$-lines one requires a Gelfand-Naimark-type theorem.

Are there any results which point in that direction, either about the calculus or positive elements in Fréchet algebras?

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    $\begingroup$ In an embarassing stroke of fortune, my google bubble just burst. I have just now found a book ("Topological Algebras with Involution" by M. Fragoulopoulou) in which my question is answered, in more general terms. Apparently, one can define something called a local $C^{*}$ algebra. This is a complete involutive topological algebra where each semi-norm $p$ is submultiplicative and fulfills $p(x)^{2}\leq p(x^{*}x)$. The cone of positive elements in such an algebra is non-empty, closed, convex and has the same characterisations as in the $C^{*}$ case. $\endgroup$
    – David Hornshaw
    Apr 16, 2016 at 8:52

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