Double centralizer algebras of certain Banach algebras

If P is a J x I matrix over an arbitrary Banach algebra A with \\P\(infinity) less than or equal to 1, then l(1) (I x J, A) with product A o B = APB is a Banach algebra which we call an l(1)-Munn algebra. In this article we study double centralizer algebras of l(1)-Munn algebras over non-unital Banach algebras. We show that if an l(1)-Munn algebra has a bounded approximate identity, then its index sets I and J are finite, its underlying algebra has a bounded approximate identity and P is regular. This result is used to give a description of double centralizers and multipliers of approximately unital l(1)-Munn algebras. Also we show that if S is a regular semigroup which admits a principal series and l(1) (S) has a bounded approximate identity, then l(1) (S) is unital and the set of idempotents of S is finite.