Discrete Semigroup Algebra

Let S be a semigroup and alpha a positive real-valued function on S such that alpha(st)<=alpha(s)alpha(t) (s,t in S). If l^1(S,alpha) is the set of all complex-valued functions f on S for which sum_(s in S)|f(s)||alpha(s)|<infty, then l^1(S,alpha) with the usual pointwise addition, scalar multiplication, the product (convolution) (f*g)(s)=sum_(tu=s)f(t)g(u) (if tu=s has no solutions, we assume (f*g)(s)=0), and with the norm ||f||=sum_(s in S)|f(s)|alpha(s) is a Banach algebra.

If alpha(s)=1, then l^1(S,alpha)=l^1(S) is called discrete semi-group algebra. Moreover if S=G is a group then l^1(S) is the discrete group algebra l^1(G).

This entry contributed by Mohammad Sal Moslehian

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Moslehian, Mohammad Sal. "Discrete Semigroup Algebra." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/DiscreteSemigroupAlgebra.html

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