Group-graded rings satisfying the strong rank condition
Group-graded rings satisfying the strong rank condition
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 Luck. In addition, we include two applications to the study of group rings and their modules.
326-338
Kropholler, Peter
0a2b4a66-9f0d-4c52-8541-3e4b2214b9f4
Lorensen, Karl
4401f837-c717-4d41-8594-a9fc4d6a39e9
1 December 2019
Kropholler, Peter
0a2b4a66-9f0d-4c52-8541-3e4b2214b9f4
Lorensen, Karl
4401f837-c717-4d41-8594-a9fc4d6a39e9
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 Luck. In addition, we include two applications to the study of group rings and their modules.
Text
SRC
- Accepted Manuscript
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Accepted/In Press date: 8 August 2019
e-pub ahead of print date: 22 August 2019
Published date: 1 December 2019
Identifiers
Local EPrints ID: 433444
URI: http://eprints.soton.ac.uk/id/eprint/433444
ISSN: 0021-8693
PURE UUID: 25f0aac5-a73c-4f20-900c-91770c78206d
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Date deposited: 22 Aug 2019 16:30
Last modified: 16 Mar 2024 08:06
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Author:
Karl Lorensen
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