Amenability and paradoxicality in semigroups and C*-algebras

We analyze the dichotomy amenable/paradoxical in the context of (discrete, countable, unital) semigroups and corresponding semigroup rings. We consider also F{\o}lner's type characterizations of amenability and give an example of a semigroup whose semigroup ring is algebraically amenable but has no F{\o}lner sequence. In the context of inverse semigroups $S$ we give a characterization of invariant measures on $S$ (in the sense of Day) in terms of two notions: $domain$ $measurability$ and $localization$. Given a unital representation of $S$ in terms of partial bijections on some set $X$ we define a natural generalization of the uniform Roe algebra of a group, which we denote by $\mathcal{R}_X$. We show that the following notions are then equivalent: (1) $X$ is domain measurable; (2) $X$ is not paradoxical; (3) $X$ satisfies the domain F{\o}lner condition; (4) there is an algebraically amenable dense *-subalgebra of $\mathcal{R}_X$; (5) $\mathcal{R}_X$ has an amenable trace; (6) $\mathcal{R}_X$ is not properly infinite and (7) $[0]\not=[1]$ in the $K_0$-group of $\mathcal{R}_X$. We also show that any tracial state on $\mathcal{R}_X$ is amenable. Moreover, taking into account the localization condition, we give several C*-algebraic characterizations of the amenability of $X$. Finally, we show that for a certain class of inverse semigroups, the quasidiagonality of $C_r^*\left(X\right)$ implies the amenability of $X$. The converse implication is false.Comment: 29 pages, minor corrections. Mistake in the statement of Proposition 4.19 from previous version corrected. Final version to appear in Journal of Functional Analysi

Coarse equivalence versus bijective coarse equivalence of expander graphs

We provide a characterization of when a coarse equivalence between coarse disjoint unions of expander graphs is close to a bijective coarse equivalence. We use this to show that if the uniform Roe algebras of coarse disjoint unions of expanders graphs are isomorphic, then the metric spaces must be bijectively coarsely equivalent

Uniform Roe algebras of uniformly locally finite metric spaces are rigid

We show that if $X$ and $Y$ are uniformly locally finite metric spaces whose uniform Roe algebras, $\mathrm{C}^*_u(X)$ and $\mathrm{C}^*_u(Y)$, are isomorphic as $\mathrm{C}^*$-algebras, then $X$ and $Y$ are coarsely equivalent metric spaces. Moreover, we show that coarse equivalence between $X$ and $Y$ is equivalent to Morita equivalence between $\mathrm{C}^*_u(X)$ and $\mathrm{C}^*_u(Y)$. As an application, we obtain that if $\Gamma$ and $\Lambda$ are finitely generated groups, then the crossed products $\ell_\infty(\Gamma)\rtimes_r\Gamma$ and $\ell_\infty(\Lambda)\rtimes_r\Lambda$ are isomorphic if and only if $\Gamma$ and $\Lambda$ are bi-Lipshitz equivalent

Inclusions of real rank zero

We introduce a notion of real rank zero for inclusions of C$^*$-algebras. After showing that our definition has many equivalent characterisations, we offer a complete description of the commutative case. We provide permanence and K-theoretic properties of inclusions of real rank zero and prove that a large class of full infinite inclusions have real rank zero.Comment: Improved the result in Theorem 5.7 and fixed the statement of Theorem 5.