Two-point DTC sets for Neufang's convolution algebra of nuclear operators

Neufang showed in [28] that the algebra N (Lp(G)) of nuclear operators with a convolution product is left but not right strongly Arens irregular for a large class of locally compact groups G. We study this algebra on discrete groups. We give representations Gamma of N (ℓp(G))** and g&d5; of LUC(G)* on B(ℓ p(G)) and show that the left topological centre condition for m ∈ N (ℓp(G))** is equivalent to Gamma(m) g&d5; (f) = g&d5; (f)Gamma(m) ∀f ∈ LUC(G)*. Using this result and an automatic normality result inspired by [21], we show that N (ℓp(G)) is left strongly Arens irregular (extending Neufang's result) and give 2-point left DTC (determining for the topological centre) sets for N (ℓp(G)) if G is abelian. We introduce a new notion of DTC in terms of representations for which we have a 2-point left set for N (ℓp(G)) for nonabelian G. We are able to lift right DTC sets (in the usual sense) for ℓ 1(G) (e.g. the 2-point set from [11]) to N (ℓp(G)).