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  • 1 Department of Mathematics, Faculty of Applied sciences Umm Al-Qura University, Makkah P.O.Box.56199, Saudi Arabia
  • 2 Department of Mathematics, Faculty of Sciences of Sfax, Tunisia
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Abstract

Let 𝔄 be a unital Banach algebra and ℜ its Jacobson radical. This paper investigates Banach algebras satisfying some chain conditions on closed ideals. In particular, it is shown that a Banach algebra 𝔄 satisfies the descending chain condition on closed left ideals then 𝔄/ℜ is finite dimensional. We also prove that a C *-algebra satisfies the ascending chain condition on left annihilators if and only if it is finite dimensional. Moreover, other auxiliary results are established.

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