Abstract
AbstractLet đ be a unital Banach algebra and â its Jacobson radical. This paper investigates Banach algebras satisfying some chain conditions on closed ideals. In particular, it is shown that a Banach algebra đ satisfies the descending chain condition on closed left ideals then đ/â 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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