Weights on Discrete Semigroups and Topological Groups

There are various types of weights scattered in the mathematics literature. Here we attempt to develop some methods to construct weights on discrete semigroups. Then we also develop methods to construct weights on topological groups.

Consider a strictly positive function ω on a semigroup (S, ∗) satisfying the following simple inequality (so-called, submultiplicativity): (s ∗ t) ≤ ω(s)ω(t) (s, t ∈ S).

Such function ω is called a weight on S. The pair (S, ω), so-called a semigroup with weight, plays a very important roll in constructing a class of Banach algebras; namely the weighted discrete semigroup algebra 1(S, ω). The Banach algebra structure of the algebra 1(S, ω) is influenced by these two simple objects S and ω. So it is important to develop methods of constructing weights on semigroups. There are various types of weights scattered in the mathematics literature. Here we attempt to develop some methods to construct weights on discrete semigroups. Then we also develop methods to construct weights on topological groups.