Finite-dimensional left ideals in some algebras associated with a locally compact group

Let G be a locally compact group, let L-1(G) be its group algebra, let M(G) be its usual measure algebra, let L-1(G)** be the second dual of L-1(G) wi th an Arens product, and let LUC(G)* be the conjugate of the space LUC(G) o f bounded, left uniformly continuous, complex-valued functions on G with an Arens-type product. We find all the finite-dimensional left ideals of thes e algebras. We deduce that such ideals exist in L-1(G) and M(G) if and only if G is compact, and in L-1(G)** (except those generated by right annihila tors of L-1(G)**) and LUC(G)* if and only if G is amenable.