MULTIPLIER HOPF-ALGEBRAS

In this paper we generalize the notion of Hopf algebra. We consider an algebra A, with or without identity, and a homomorphism DELTA from A to the multiplier algebra M(A x A) of A x A . We impose certain conditions on DELTA (such as coassociativity). Then we call the pair (A, DELTA) a multiplier Hopf algebra. The motivating example is the case where A is the algebra of complex, finitely supported functions on a group G and where (DELTAf)(s, t) = f(st) with s, t is-an-element-of G and f is-an-element-of A. We prove the existence of a counit and an antipode. If A has an identity, we have a usual Hopf algebra. We also consider the case where A is a *-algebra. Then we show that (a large enough) subspace of the dual space can also be made into a *-algebra.