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The algebras A,(G) of elements in L₁ (G) whose Fourier transforms belong to L,(Ĝ) and the multipliers for these algebras have been studied by various authors including Larsen, Liu and Wang [7], Figà-Talamanca and Gaudry [3], Reiter [10] and Martin and Yap [9]. The purpose of this note is to study the algebras A,(1) of elements in L₁ (I) whose Gelfand transforms belong to L,(Î) and the multipliers for these algebras, where I is the locally compact idempotent commutative topological semigroup consisting of the open interval (0,∞ o) of real numbers from 0 to co equipped with the usual topology and max. multiplication and Î is the maximal ideal space of L₁ (I). Whereas the algebras A,(G) are similar to the group algebra L₁ (G) in great many ways, the algebras A,(1) are dissimilar to the order convolution algebra L (I). In particular we shall see that the maximal ideal space▲(A,(I)) of A „(1) is not the same as that of L₁ (I). Moreover, the algebra of multipliers of A ‚(1) properly contains the algebra of multipliers of L (I). We establish below our notations and then proceed to describe the results.

Let M (I) denote the Banach algebra of all finite regular Borel measures on I under the order convolution product denoted by and total variation norm. Then the Banach space L₁ (I) of all measures in M (I) which are absolutely continuous with respect to the Lebesgue measure on I becomes a commutative semisimple Banach algebra in the inherited product*. More specifically, for fƒ, g Є L₁ (1),