Abstract

Let G be a non-compact locally compact group with a continuous submultiplicative weight function ω such that ω(e) = 1 and ω is diagonally bounded with bound K ≥ 1. When G is σ-compact, we show that [K] + 1 many points in the spectrum of LUC(ω^-1) are enough to determine the topological centre of LUC(ω^-1)^⁎ and that [K] + 2 many points in the spectrum of L^∞(ω^-1) are enough to determine the topological centre of L¹(ω)^⁎⁎ when G is in addition a SIN-group. We deduce that the topological centre of LUC(ω^-1)^⁎ is the weighted measure algebra M(ω) and that of C₀(ω^-1)^┴ is trivial for any locally compact group. The topological centre of L¹(ω)^⁎⁎ is L¹(ω) and that of L^∞₀(ω)^┴ is trivial for any non-compact locally compact SIN-group. The same techniques apply and lead to similar results when G is a weakly cancellative right cancellative discrete semigroup.