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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...
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