Strict comparison holds in the uniform Roe algebra of any countable discrete amenable group, with an additional strong AH result for the crossed product by the universal minimal set.
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Every free continuous action of a countably infinite elementary amenable group on a finite-dimensional compact metrizable space is almost finite, with the consequence that minimal crossed products are Z-stable and Elliott-classifiable.
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Strict comparison holds in the uniform Roe algebra of a discrete amenable group
Strict comparison holds in the uniform Roe algebra of any countable discrete amenable group, with an additional strong AH result for the crossed product by the universal minimal set.
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Elementary amenability and almost finiteness
Every free continuous action of a countably infinite elementary amenable group on a finite-dimensional compact metrizable space is almost finite, with the consequence that minimal crossed products are Z-stable and Elliott-classifiable.