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

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

If the answer is negative,

2- Does there exist any characterization of Banach *-algebras whose involution is continuous (isometry)?!


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 $A$

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