Projectivity of Some Banach Right Modules over the Group Algebra ℓ1(G)

Let G be a locally compact group, \({\cal B}({L^2}(G))\) be the space of all bounded linear operators on L²(G), and \(({\cal T}({L^2}(G)), \ast)\) be the Banach algebra of trace class operators on L²(G). In this paper, we focus on some Banach right submodules of \({\cal B}({L^2}(G))\) over the convolution algebras \(({\cal T}({L^2}(G)), \ast)\) and (L¹(G),*). We will see that if the locally compact group G is discrete, then the Banach right ℓ¹(G)-module structures of them are derived from their Banach right \({\cal T}({\ell ^2}(G))\)-module structures. We also study the projectivity of these Banach right ℓ¹(G)-modules.