Group-graded rings satisfying the strong rank condition

2019/01/28 by Kropholler, Peter, Lorensen, Karl · 2 citations
#16P99 #16W50 #20F65 #43A07 #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1901.10001

Abstract

A ring R satisfies the \it 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.

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