Abstract
If P is a J×I matrix over an arbitrary Banach algebra \({\cal A}\) with \(\Vert P\Vert_\infty \le 1\), then \(\ell^1 (I \times J, {\cal J}\) with product A○B = APB is a Banach algebra which we call an l1-Munn algebra. In this article we study double centralizer algebras of l1-Munn algebras over non-unital Banach algebras. We show that if an l1-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 l1-Munn algebras. Also we show that if S is a regular semigroup which admits a principal series and l1(S) has a bounded approximate identity, then l1(S) is unital and the set of idempotents of S is finite.
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