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A locally convex $C^*$ algebra is a locally convex topological vector space $A$ whose topology is generated by a familly of complete $C^*$ semi norm and $A$ is a $*$ algebra. Moreover all algebra operations and involution are continuous operators(A complete $C^*$ semi norm is a semi norm which satisfies $|xx^*|=|x|^2$ and is a complete semi norm). Locally convex Banach algebras is introduced in "Banach and locally convex algebras" by A.Ya. Helemeskii

Question: Is there a locally convex $C^*$ algebraic structure on the disk algebra $A(\mathbb{D})$?

The disk algebra is the Banach algebra of all holomorphic functions on the interior of the unit disk with continuous extention the boundary of the disk.

A motivation for this question is the following post:

A locally convex $C^*$ algebra without zero divisor

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    $\begingroup$ It is rather unusual to have a complete seminorm $p$ on Fréchet space $X$ because then the the quotient map $X\to (X,p)/\mathrm{kern}(p)$ would be open (because the quotient is a Banach space). Such Fréchet spaces are called quojections and very close to products of Banach spaces. $\endgroup$
    – Jochen Wengenroth
    Jun 23 at 16:55
  • $\begingroup$ @JochenWengenroth Thank you very much for your comments and your attention to my question. $\endgroup$
    – Ali Taghavi
    2 days ago
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    $\begingroup$ Ali, I must say that I don't understand what is meant by "complete seminorm". $\endgroup$
    – Sergei Akbarov
    2 days ago
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    $\begingroup$ @SergeiAkbarov Sergei I mean that the quotint space is complete. $\endgroup$
    – Ali Taghavi
    yesterday
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    $\begingroup$ Ah, OK! $\phantom{-----}$ $\endgroup$
    – Sergei Akbarov
    yesterday

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