Isometric multipliers and isometric isomorphisms of $l\sb{1}(S)$

Abstract

Let S be a commutative semigroup and 0(S) the multiplier semigroup of 5. It is shown that T is an isometric multiplier of lx(S) if and only if there exists an invertible element o S Í2(S) and a complex number X of unit modulus such that T(a) = \^,xes a(x)Sa,x, for each a-2,es «M*x e'l(S)-Also, if 5, and S2 are commutative semigroups, and L is an isometric isomorphism of /|(5,) into/,(S2), then it is proved that there exist a semicharacter x, IxMI = 1 Ior all x £ Sx, and an isomorphism /' of S, onto S2 such that L(a) = 2 x(*)«(*)*f(x) for each a = 2xes, a(x)sx e 'i(^i)