Amenability of Restricted Semigroup Algebras
Mohammad Mehran, Massoud Amini, Ali R. Medghalchi, Ali Ebadian · 2010
In 1972, B.E. Johnson proved that for every discrete group G, (G) is amenable as a Banach algebra if and only if G is amenable as a group [9]. When S is a commutative semigroup, (S) is amenable if and only if S is a finite semilattice of abelian (and hence amenable) groups [7]. When S is a cancellative semigroup with identity, it is amenable if and only if S is an amenable group [6]. In 1978, J. Duncan and I. Namioka showed that if S is an arbitrary inverse semigroup with finite set of idempotents E(S), then (S) is amenable if and only if each maximal group of S is amenable [4]. Also, they showed that (S) fails to be amenable if E(S) is infinite, for Eunitary semigroups. In 1990, J. Duncan and A.L.T. Paterson completed the story for inverse semigroup by showing that the above result holds without the restriction of S being E-unitary [5]. Recently, G.K. Dales, A.T.-M. Lau and D. Strauss have shown that for an arbitrary semigroup S, (S) is amenable if and only if S is ’built up from amenable groups’ [3, Theorem 10.12]. They used the methods of [4]. We apply some of the above results to the semigroup algebra (Sr), where Sr is the restricted 0-semigroup associated to an inverse semigroup S [1], to prove similar results about the restricted semigroup algebra r(S), introduced by the second author and A.R. Medghalchi in [1]. We show that the restricted