Module maps on duals of Banach algebras and topological centre problems

We study various spaces of module maps on the dual of a Banach algebra A, and relate them to topological centres. We introduce an auxiliary topological centre Zt(⁎〈A⁎A〉)⋄ for the left quotient Banach algebra ⁎〈A⁎A〉 of A⁎⁎. Our results indicate that Zt(⁎〈A⁎A〉)⋄ is indispensable for investigating properties of module maps over A⁎ and for understanding some asymmetry phenomena in topological centre problems as well as the interrelationships between different Arens irregularity properties. For the class of Banach algebras of type (M) introduced recently by the authors, we show that strong Arens irregularity can be expressed both in terms of automatic normality of A⁎⁎-module maps on A⁎ and through certain commutation relations. This in particular generalizes the earlier work on group algebras by Ghahramani and McClure (1992) [13], and by Ghahramani and Lau (1997) [12], . We link a module map property over A⁎ to the space WAP(A) of weakly almost periodic functionals on A, generalizing a result by Lau and Ülger (1996) [34] for Banach algebras with a bounded approximate identity. We also show that for a locally compact quantum group G, the quotient strong Arens irregularity of L1(G) can be obtained from that of M(G) and can be characterized via the canonical C0(G)-module structure on LUC⁎(G).