fa.functional analysis – A locally convex $C^*$ algebraic structure on the disk algebra Answer

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fa.functional analysis – 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 family of complete $C^*$ semi normal 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).

Questions: 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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