Resumen de Unital Banach algebras and their subalgebras

Miao Tianxuan

• Let A be a Banach algebra with a bounded approximate identity. We prove that if A is not unital, then there is a nonunital subalgebra B of A with a sequential bounded approximate identity. It follows that A must be unital if A is weakly sequentially complete and B** under the first Arens multiplication has a unique right identity for every subalgebra B of A with a sequential bounded approximate identity. As a consequence, we prove a result of Ülger that if A is both weakly sequentially complete and Arens regular, then A must be unital.