On the radical banach algebras related to semigroup algebras

Let S be a compactly cancellative foundation semigroup with identity. It is well-known that L¥ 0 (S;Ma(S)) can be equipped with a multiplication that extends the original multiplication on Ma(S) and makes L¥ 0 (S;Ma(S)) a Banach algebra. In this paper, among the other things, it is shown that if S is a nondiscrete compactly cancellative foundation semigroup with an identity, then the radical of L¥ 0 (S;Ma(S)) is infinitedimensional. 2010 Mathematics Subject Classification: Primary: 43A05; Secondary: 46H10