• Spaces of vector-valued integrable functions;
  • strict topology;
  • generalized group algebra;
  • locally compact group;
  • semi-topological algebra;
  • MSC (2010) Primary: 46A03;
  • 46A70;
  • Secondary: 46E40;
  • 43A20


Let E be a Banach space, Ω a locally compact space, and μ a positive Radon measure on Ω. In this paper, through extending to Lebesgue-Bochner spaces, we show that the topology β1 introduced by Singh is a type of strict topology. We then investigate various properties of this locally convex topology. In particular, we show that the strong dual of L1(μ, E) can be identified with a Banach space. We also show that the topology β1 is a metrizable, barrelled or bornological space if and only if Ω is compact. Finally, we consider the generalized group algebra equation image under certain natural locally convex topologies. As an application of our results, we prove that equation image under the topology β1 is a complete semi-topological algebra.