# 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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