Fell-Bündel und verallgemeinerte L1-algebren

The relation between cross-sectional algebras of homogeneous Banach-1-algebraic bundles in the sense of Fell 5] and generalized $\text{L}$¹-algebras, as defined in slightly different ways by Leptin 7], Busby and Smith 2], and others, has been studied by Busby in 3]. We give an extension of his result, using a different method for obtaining topological group extensions. Instead of first constructing the abstract group extension from the given factor system and then topologizing it, we work in a natural topological setting and define a topological group which turns out to be the group extension belonging to the given factor system. As a consequence we obtain (without separability assumptions) that for any measurable factor system of a locally compact group with values in some other locally compact group the corresponding abstract group extension can be topologized to give a topological (and hence locally compact) group extension.