# Continuity of the involution in Banach *-algebras

My question is concerned with the involution in Banach *-algebras.

1- Should the involution be assumed continuous in every Banach *-algebra?

For 1), this is § 36 (pp. 190) of Bonsall & Duncan: The following are equivalent for a Banach *-algebra $$A$$, and the set $$\text{Sym}(A')$$ of all continuous self-adjoint linear functionals on $$A$$
• The linear involution $$f \mapsto f^*$$ is continuous on $$A$$
• $$\text{Sym}(A')$$ is separating on $$A$$
• $$\text{Sym}(A)$$ is closed on $$A$$