Amenability and uniform Roe algebras

Abstract

Amenability for groups can be extended to metric spaces, algebras over commutative fields and $C^{\ast }$-algebras by adapting the notion of Følner nets. In the present article we investigate the close ties among these extensions and show that these three pictures unify in the context of the uniform Roe algebra $C_u^{\ast }(X)$ over a metric space $(X,d)$ with bounded geometry. In particular, we show that the following conditions are equivalent: (1) $(X,d)$ is amenable; (2) the translation algebra generating $C_u^{\ast }(X)$ is algebraically amenable (3) $C_u^{\ast }(X)$ has a tracial state; (4) $C_u^{\ast }(X)$ is not properly infinite; (5) $[1{]}_0\ne [0{]}_0$ in the $K_0$-group $K_0(C_u^{\ast }(X))$; (6) $C_u^{\ast }(X)$ does not contain the Leavitt algebra as a unital $\ast$-subalgebra; (7) $C_u^{\ast }(X)$ is a Følner $C^{\ast }$-algebra in the sense that it admits a net of unital completely positive maps into matrices which is asymptotically multiplicative in the normalized trace norm. We also show that every possible tracial state of the uniform Roe algebra $C_u^{\ast }(X)$ is amenable.

arXiv commentfinal version; Remark 4.20 added solving partially Problem 4.19; typos corrected; references updated; 30 pages

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Cited by in corpus (10)

• Ideal structure and pure infiniteness of ample groupoid $C^{\ast }$-algebras
• Rigidity of ${\ell }^p$ Roe-type algebras
• A quasi-local characterisation of $L^p$-Roe algebras
• The type semigroup, comparison and almost finiteness for ample groupoids
• Amenability and paradoxicality in semigroups and C*-algebras
• The uniform Roe algebra of an inverse semigroup
• Quasi-local Algebras and Asymptotic Expanders
• Structure and $K$-theory of ${\ell }^p$ uniform Roe algebras
• On topological obstructions to the existence of non-periodic Wannier bases
• A note on commutation relations and finite dimensional approximations