Summability and invariant means on semigroups

This thesis consists of two parts. In the first part, we study summability in left amenable semigroups. More explicitly, various summability methods defined by matrices are considered. Necessary and (or) sufficient conditions are given for matrices to be regular, almost regular, Schur, almost Schur, strongly regular and almost strongly regular, generalizing those of O. Toeplitz, J. P. King, J. Schur, G. G. Lorentz and P. Schaefer for the semigroup of additive positive integers. The theorems are of interest even for the semigroup of multiplicative positive integers. Let S be a topological semigroup which is amenable as a discrete semigroup. Denote by LUC(S) the set of bounded real-valued left uniformly continuous functions on S. It is shown by E. Granirer that if S is a separable topological group which is amenable as a discrete group and has a certain property (B) then LUC(S) has "many" left invariant means. In the second part of this thesis, we extend this result to certain topological subsemigroups of a topological group. In particular, we show that if S is a separable closed non-compact subsemigroup of a locally compact group which is amenable as a discrete semigroup then LUC(S) has "many" left invariant means. Finally, an example is given to show that this result cannot be extended to every topological semigroup.Science, Faculty ofMathematics, Department ofGraduat