We study the family of all not necessarily complete algebra norms on a semisimple Banach algebra as a partially ordered set and investigate the existence and properties of minimal elements.
Agrafeuil and Zarrabi in [1] characterized all closed ideals with at most countable hull in a unital Banach algebra embedded in the classical disc algebra and satisfying certain conditions ((H1), (H2), (H3)), and the analytic Ditkin condition. We modify Ditkin’s condition and show that analogous result is true for a wider class of algebras. This is an extension of the result obtained in [1].
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