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}, }

Downloads: 0