Continuous Bounded Cohomology of Topological Semigroups

We define a cohomology theory for topological semigroups, with seminormed cohomology groups. Our theory can be considered as a topological version of bounded cohomology of discrete semigroups. 1. Definition of the cohomology Bounded cohomology was first defined for discrete groups by F. Trauber and then for topological spaces by M. Gromov [6]. In this paper we establish a topological bounded cohomology theory for topological semigroups, using continuous bounded cocycles. For any set X, B(X) denotes the Banach space of all bounded complex (C) valued maps onX with the uniform norm. IfX has a topology, then C(X) ⊂ B(X) denotes the Banach subspace of continuous maps. By a topological semigroup we mean a semigroup S with a topology such that the multiplication S × S −→ S is jointly continuous. Let S be a semigroup. Let C b (S) = C, and for n ≥ 1, let C n b (S) = B(S). The elements of C b (S) are called bounded cochains of the semigroup S. Let δ : C b (S) −→ C 1 b (S) be the zero linear map and for n ≥ 1, define the bounded linear map δ : C b (S) −→ C n+1 b (S) by δ(f)(s1, · · · , sn+1) = f(s2, · · · , sn+1)