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arXiv · 1901.10001

Group-graded rings satisfying the strong rank condition

Peter Kropholler·Karl Lorensen
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Abstract

A ring $R$ satisfies the {\it strong rank condition} (SRC) if, for every natural number $n$, the free $R$-submodules of $R^n$ all have rank $\leq n$. Let $G$ be a group and $R$ a ring strongly graded by $G$ such that the base ring $R_1$ is a domain. Using an argument originated by Laurent Bartholdi for studying cellular automata, we prove that $R$ satisfies SRC if and only if $R_1$ 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 L\"uck. In addition, we include two applications to the study of group rings and their modules.

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Peter Kropholler, Karl Lorensen. 2019-01-28. Group-graded rings satisfying the strong rank condition. https://arxiv.org/abs/1901.10001

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