Weighted Lp-conjecture for locally compact groups

Let G be a locally compact group, ω be a weight function on G and 1 < p < ∞. Here, we give a sufficient condition for that the weighted Lp-space Lp(G, ω) is a Banach algebra. Also, we get some necessary conditions on G and the weight function ω for Lp(G, ω) to be a Banach algebra. As a consequence, we show that if G is abelian and Lp(G, ω) is a Banach algebra, then G is σ-compact.