The Functions Operating on Certain Algebras of Multipliers

In this note, we announce a new result concerning functions operating on multiplier algebras. We begin by introducing the following notation. Let G be a LCA group with dual group I\ M(G) will denote the algebra of finite, regular Borel measures on G. Let M0(G) = {M E M(G)\ jl vanishes at <*> on T}. If 1 < p < °°, let M (G) denote the class of multiplier transformations on L (G). If T G Mp(G), f will be the unique function in L^ÇT) so that T(f)~ = Tf, for all integrable simple functions/. Finally, we writeC0Mp(G) = {TEMp(G)\ T is continuous and vanishes at °° on T}. Suppose that G is nondiscrete. It is well known that only entire functions operate on the Banach algebra M(G) [3, Chapter 6]. This result was strengthened in [1]. There, Igari showed that only entire functions operate from M(G) into the algebra Mp(G), 1 < p < <*>, p =£ 2. In [4], Varopoulos showed that for compact G, only entire functions operate on M0(G). We have the following theorems, which, in a sense, may be viewed as the L analogues of the aforementioned result of Varopoulos.