Finite dimensionality in socle of Banach algebras (Q1067557)

It is shown that if the socle soc(A) of a semisimple Banach algebra A is norm-closed, then soc(A) is already finite dimensional. This is an extension of Al-Moajil's result which states that if A is a semi-simple Banach algebra such that \(lan(comp(A))=\{0\}\) and soc(A) is norm-closed, then A is finite dimensional. Here comp(A) and lan(comp(A)) denote respectively the compactrum of A and the left annihilator of comp(A). The proof makes use of the above Al-Moajil's result and the fact that if A is a semisimple Banach algebra with socle soc(A) and min(A) is the set of all minimal idempotents of A, then \(lan(comp(A))=lan(soc(A))=lan(\min (A))=ran(\min (A))=ran(comp(A))\), where ran(X) denotes the right annihilator of X.