On closed Lie ideals and center of generalized group algebras

Abstract

For any locally compact group G and any Banach algebra A, a characterization of the closed Lie ideals of the generalized group algebra L1(G,A) is obtained in terms of left and right actions by G and A. In addition, when A is unital and G is an [SIN] group, we show that the center of L1(G,A) is precisely the collection of all center valued functions which are constant on the conjugacy classes of G. As an application, we establish that Z(L1(G)⊗γA)=Z(L1(G))⊗γZ(A), for a class of groups and Banach algebras. And, prior to these, for any finite group G, the Lie ideals of the group algebra C[G] are identified in terms of some canonical spaces determined by the irreducible characters of G.