ON THE SPECTRUM OF L(INFINITY)(G)

We investigate the algebraic structure of the spectrum OMEGA of L(infi nity)(G) for a locally compact group G. In contrast to the compact and discrete cases, when G has neither of these properties, OMEGA is neve r a semigroup. For sigma-compact G we determine exactly when the produ ct of two elements of OMEGA is in OMEGA, but we present an example whi ch suggests that for general groups the underlying set theory may have an effect. Our principal tool, which has independent interest, is a t opological structure theorem for the LC-compactification of an arbitra ry locally compact group.