The kh-socle of a commutative semisimple Banach algebra

Abstract

Let $\mathcal{A}$ be a commutative complex semisimple Banach algebra. Denote by ${\rm kh}({\rm soc}(\mathcal{A}))$ the kernel of the hull of the socle of $\mathcal{A}$. In this work we give some new characterizations of this ideal in terms of minimal idempotents in $\mathcal{A}$. This allows us to show that a "result" from Riesz theory in commutative Banach algebras is not true.

Keywords

• commutative banach algebra
• socle
• kh-socle
• inessential element