The \(\mathrm{kh}\)-socle of a commutative semisimple Banach algebra. (English) Zbl 1499.46108

Hadder, Youness

Math. Bohem. 145, No. 4, 387-399 (2020).

Summary: Let \(\mathcal{A}\) be a commutative complex semisimple Banach algebra. Denote by \(\mathrm{kh}(\mathrm{soc}(\mathcal{A}))\) 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.

MSC:

46J05	General theory of commutative topological algebras
46J20	Ideals, maximal ideals, boundaries
47A10	Spectrum, resolvent

Keywords:

commutative Banach algebra; socle; \(\mathrm{kh}\)-socle; inessential element

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