# A locally convex $C^*$ algebraic structure on the disk algebra

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

• 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. Jun 23 at 16:55
• @JochenWengenroth Thank you very much for your comments and your attention to my question. 2 days ago
• Ali, I must say that I don't understand what is meant by "complete seminorm". 2 days ago
• @SergeiAkbarov Sergei I mean that the quotint space is complete. yesterday
• Ah, OK! $\phantom{-----}$ yesterday