Resumen de Injective Convolution Operators on l8(G) are Surjective

Yemon Choi

• Let G be a discrete group and let f ? l1(G). We observe that if the natural convolution operator ?f: l8(G)? l8(G) is injective, then f is invertible in l1(G). Our proof simplifies and generalizes calculations in a preprint of Deninger and Schmidt by appealing to the direct finiteness of the algebra l1(G). We give simple examples to show that in general one cannot replace l8 with lp, 1 = p < 8, nor with L8(G) for nondiscrete G. Finally, we consider the problem of extending the main result to the case of weighted convolution operators on G, and give some partial results.