Group-Graded Rings Satisfying the Strong Rank Condition

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Date
2019-08-16MFO Scientific Program
Research in Pairs 2015Series
Oberwolfach Preprints;2019,22Author
Kropholler, Peter H.
Lorensen, Karl
Metadata
Show full item recordOWP-2019-22
Abstract
A ring $R$ satisfies the $\textit{strong rank condition}$ (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 Lück. In addition, we include two applications to the study of group rings and their modules.