Abstract
In this work, we introduce and study the properties of topologically Noetherian Banach algebras. In particular, we prove, if every prime closed ideal of a commutative Banach algebra A is maximal, t...
Highlights
Grauert and Remmert (1971) proved if every closed ideal in a commutative Banach algebra A is finitely generated, A is finite dimensional. Sinclair and Tullo (1974) obtained a non-commutative version of this result. Ferreira and Tomassini (1978) improved Grauert and Remmert’s result by showing that the statement is true if one replaces “closed ideals” by “maximal ideals” in the Shilov boundary of A
A closed ideal I of a commutative Banach algebra A is called primary if the conditions ab ∊ I and a ∉ I together imply bn ∊ I, for some positive integer n
It is clear that any left Noetherian and any simple Banach algebra is topologically Noetherian
Summary
Grauert and Remmert (1971) proved if every closed ideal in a commutative Banach algebra A is finitely generated, A is finite dimensional. Sinclair and Tullo (1974) obtained a non-commutative version of this result. Ferreira and Tomassini (1978) improved Grauert and Remmert’s result by showing that the statement is true if one replaces “closed ideals” by “maximal ideals” in the Shilov boundary of A. Grauert and Remmert (1971) proved if every closed ideal in a commutative Banach algebra A is finitely generated, A is finite dimensional. I is said to be irreducible if it is not a finite intersection of closed ideals of A properly containing I, otherwise, I is termed reducible. A closed ideal I of a commutative Banach algebra A is called primary if the conditions ab ∊ I and a ∉ I together imply bn ∊ I, for some positive integer n. A left ideal I of A is said to be topologically finitely generated (t.f.g.) if there exist x1, x2, ..., xn ∊ I such that Ī = < x1, x2, ... Of closed left ideals of A, there exists n∊ Z+ such that. It is clear that any left Noetherian and any simple Banach algebra is topologically Noetherian
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