Group-Graded Rings Satisfying the Strong Rank Condition

Abstract

A ring R satisfies the strong rank condition (SRC) if, for every natural number n, the free R-submodules of Rⁿ all have rank ≤n. Let G be a group and R a ring strongly graded by G such that the base ring R₁ is a domain. Using an argument originated by Laurent Bartholdi for studying cellular automata, we prove that R satisfies SRC if and only if R₁ satisfies SRC and G is amenable. The special case of this result for group rings allows us to prove a characterization of amenability involving the group von Neumann algebra that was conjectured by Wolfgang Lück. In addition, we include two applications to the study of group rings and their modules.