proceedings of the american mathematical society Volume 117, Number 3, March 1993 MULTIPLIERS ON SUBSEMIGROUPS OF THE REAL LINE HUNG T. DINH (Communicated by Palle E. T. Jorgensen) Abstract. We show that multipliers on subsemigroups of the real line can be extended to multipliers on groups. Multipliers on subsemigroups of the real line arise from the index theory of semigroups of endomorphisms of type I factors [1, 2, 4, 5]. Although multipliers on groups have been extensively studied [7], little is known about multipliers on semigroups. Let G be a subgroup of R and G+ be the semigroup of nonnegative elements of G. A multiplier of G+ (resp. G) is a Borel-measurable function f:G+x G+ - T (resp. f:GxG-+T) satisfying m f(r,s)f(r + s,t) = f(s,t)f(r,s + t), U f(t,0) = f(0,t) = l for every r, s, t e G+ (resp. G). We say that two multipliers f and f2 on G+ are similar if there is a measurable function a: G+ —> T satisfying (2» '*•<>=»*<*•<> for every s, t e G+ . Let M'(G+) be the set of all multipliers on G+ . Then M'{G+) is an abelian group under pointwise multiplication. We say that a multiplier / on G+ is trivial or exact if it is of the form (3) f(s,t) = a(s + t)/a(s)a(t) for some measurable a: G+ —► T. The set of all trivial multipliers form a subgroup Mq(G+) of M'(G+). Finally, the multiplier group is defined by M(G+) = M'(G+)/M()(G+). The group M(G) is defined similarly [7]. We will show that every multiplier on the semigroup G+ can be extended to a multiplier on the whole group G, thus reducing the study of semigroup Received by the editors April 10, 1991 and, in revised form, July 1, 1991. 1991 Mathematics Subject Classification. Primary 46L55, 47D03; Secondary 20C25. © 1993 American Mathematical Society 0002-9939/93 $1.00+ $.25 per page 783 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use