On rigidity of Roe algebras. Špakula, J. & Willett, R. Advances in Mathematics, 249:289–310, December, 2013. Paper doi abstract bibtex Roe algebras are C ⁎ -algebras built using large scale (or ‘coarse’) aspects of a metric space ( X , d ) . In the special case that X = Γ is a finitely generated group and d is a word metric, the simplest Roe algebra associated to ( Γ , d ) is isomorphic to the crossed product C ⁎ -algebra l ∞ ( Γ ) ⋊ r Γ . Roe algebras are coarse invariants, meaning that if X and Y are coarsely equivalent metric spaces, then their Roe algebras are isomorphic. Motivated in part by the coarse Baum–Connes conjecture, we show that the converse statement is true for a very large classes of spaces. This can be thought of as a ‘ C ⁎ -rigidity result’: it shows that the Roe algebra construction preserves a large amount of information about the space, and is thus surprisingly ‘rigid’. As an example of our results, in the group case we have that if Γ and Λ are finitely generated amenable, hyperbolic, or linear, groups such that the crossed products l ∞ ( Γ ) ⋊ r Γ and l ∞ ( Λ ) ⋊ r Λ are isomorphic, then Γ and Λ are quasi-isometric.
@article{spakula_rigidity_2013,
title = {On rigidity of {Roe} algebras},
volume = {249},
issn = {0001-8708},
url = {http://www.sciencedirect.com/science/article/pii/S0001870813003320},
doi = {10.1016/j.aim.2013.09.006},
abstract = {Roe algebras are C ⁎ -algebras built using large scale (or ‘coarse’) aspects of a metric space ( X , d ) . In the special case that X = Γ is a finitely generated group and d is a word metric, the simplest Roe algebra associated to ( Γ , d ) is isomorphic to the crossed product C ⁎ -algebra l ∞ ( Γ ) ⋊ r Γ .
Roe algebras are coarse invariants, meaning that if X and Y are coarsely equivalent metric spaces, then their Roe algebras are isomorphic. Motivated in part by the coarse Baum–Connes conjecture, we show that the converse statement is true for a very large classes of spaces. This can be thought of as a ‘ C ⁎ -rigidity result’: it shows that the Roe algebra construction preserves a large amount of information about the space, and is thus surprisingly ‘rigid’.
As an example of our results, in the group case we have that if Γ and Λ are finitely generated amenable, hyperbolic, or linear, groups such that the crossed products l ∞ ( Γ ) ⋊ r Γ and l ∞ ( Λ ) ⋊ r Λ are isomorphic, then Γ and Λ are quasi-isometric.},
urldate = {2016-07-19},
journal = {Advances in Mathematics},
author = {Špakula, Ján and Willett, Rufus},
month = dec,
year = {2013},
keywords = {Coarse Baum–Connes conjecture, Coarse geometry},
pages = {289--310},
}
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