Some Results in the Topological Centers of Module Actions

Let B be a Banach Abimodule and let ( ) ** ** Z A B and ( ) ** ** Z B A be the topological centers of the left module action : ABB and the right module action : , r  B AB respectively. We investigate the relation between topological center of ** A , ( ) ** 1 Z A with respect to the first Arens product and topological centers of module actions ( ) ** ** Z A B and ( ) ** ** Z B A . On the other hand, if A has Mazure property and ** B has the left A  factorization ** , then Z B B A ( ) = ** ** . For a locally compact non-compact group G with compact covering number card(G), we have ( ( ) ) = ( ) 1 ** 1 ( )** Z L G L G M G and ( ( ) ) = ( ) ** 1( )** Z M G M G L G. .