S là một Hausdorff tại địa phương-nửa nhóm topo nhỏ gọn, và M (S) là các đại số Banach của tất cả các biện pháp Borel thường xuyên bị giới hạn S. Hãy Ma (S) là không gian của tất cả các biện pháp μ ∈ M (S) như vậy mà cả hai bản đồ x → δx * | μ | và x → μ * δx từ S vào M (S) là yếu liên tục. | Vietnam Journal of Mathematics 35 1 2007 33-41 Viet n a m J 0 u r n a I of MATHEMATICS VAST 2007 Amenable Locally Compact Foundation Semigroups Ali Ghaffari Department of Mathematics Semnan University Semnan Iran Received January 11 2006 Revised October 24 2006 Abstract. Let S be a locally compact Hausdorff topological semigroup and M S be the Banach algebra of all bounded regular Borel measures on S. Let Ma S be the space of all measures p G M S such that both mapping x ỗx p and x p ỗx from S into M S are weakly continuous. In this paper we present a few results in the theory of amenable foundation semigroups. A number of theorems are established about left invariant mean of a foundation semigroup. In particular we establish theorems which show that Ma S has a left invariant mean. Some results were previously known for groups. 2000 Mathematics Subject Classification 22A20 43A60. Keywords Banach algebras locally compact semigroup topologically left invariant mean fixed point 1. Introduction Let S be a locally compact Hausdorff topological semigroup and M S the Banach algebra of all bounded regular Borel measures on S with total variation norm and convolution p V p V G M S as multiplication where y fdp V Ị Ị f xy dp x dv y Ị Ị f xy dv y dp x for f G Co S the space of all continuous functions on S which vanish at infinity. see for example 5 11 or 13 . Let Mo S be the set of all probability measures 34 Ali Ghaffari in M S . Let Ma S 1 5 12 denote the space of all measures p G M S such that both mappings x 5x p and x p 5x from S into M S are weakly continuous. A semigroup S is called a foundation semigroup if pi slippy p G Ma S is dense in S. In this paper we may assume that S is a foundation locally compact Hausdorff topological semigroup with identity e. Note that Ma S is a closed two-sided L-ideal of M S 5 . We also note that for p G Ma S both mappings x ỗx p and x p ỗx from S into M S are norm continuous 5 . It is known that Ma S admits a bounded approximate .
