An textit{algebraic} action of a discrete group $Gamma $ is a homomorphism from $Gamma $ to the group of continuous automorphisms of a compact abelian group $X$. By duality, such an action of $Gamma $ is determined by a module $M=widehat{X}$ over the
integer group ring $mathbb{Z}Gamma $ of $Gamma $. The simplest examples of such modules are of the form $M=mathbb{Z}Gamma /mathbb{Z}Gamma f$ with $fin mathbb{Z}Gamma $; the corresponding algebraic action is the textit{principal algebraic $Gamma $-action} $alpha _f$ defined by $f$. In this note we prove the following extensions of results by Hayes cite{Hayes} on ergodicity of principal algebraic actions: If $Gamma $ is a countably infinite discrete group which is not virtually cyclic, and if $finmathbb{Z}Gamma $ satisfies that right multiplication by $f$ on $ell ^2(Gamma ,mathbb{R})$ is injective, then the principal $Gamma $-action $alpha _f$ is ergodic (Theorem ref{t:ergodic2}). If $Gamma $ contains a finitely generated subgroup with a single end (e.g. a finitely generated amenable subgroup which is not virtually cyclic), or an infinite nonamenable subgroup with vanishing first $ell ^2$-Betti number (e.g., an infinite property $T$ subgroup), the injectivity condition on $f$ can be replaced by the weaker hypothesis that $f$ is not a right zero-divisor in $mathbb{Z}Gamma $ (Theorem ref{t:ergodic1}). Finally, if $Gamma $ is torsion-free, not virtually cyclic, and satisfies Linnells textit{analytic zero-divisor conjecture}, then $alpha _f$ is ergodic for every $fin mathbb{Z}Gamma $ (Remark ref{r:analytic zero divisor}).
We introduce an invariant, called mean rank, for any module M of the integral group ring of a discrete amenable group $Gamma$, as an analogue of the rank of an abelian group. It is shown that the mean dimension of the induced $Gamma$-action on the Po
ntryagin dual of M, the mean rank of M, and the von Neumann-Luck rank of M all coincide. As applications, we establish an addition formula for mean dimension of algebraic actions, prove the analogue of the Pontryagin-Schnirelmnn theorem for algebraic actions, and show that for elementary amenable groups with an upper bound on the orders of finite subgroups, algebraic actions with zero mean dimension are inverse limits of finite entropy actions.
We show that any Lipschitz projection-valued function p on a connected closed Riemannian manifold can be approximated uniformly by smooth projection-valued functions q with Lipschitz constant close to that of p. This answers a question of Rieffel.
We introduce mean dimensions for continuous actions of countable sofic groups on compact metrizable spaces. These generalize the Gromov-Lindenstrauss-Weiss mean dimensions for actions of countable amenable groups, and are useful for distinguishing co
ntinuous actions of countable sofic groups with infinite entropy.
We show that, for countable sofic groups, a Bernoulli action with infinite entropy base has infinite entropy with respect to every sofic approximation sequence. This builds on the work of Lewis Bowen in the case of finite entropy base and completes t
he computation of measure entropy for Bernoulli actions over countable sofic groups. One consequence is that such a Bernoulli action fails to have a generating countable partition with finite entropy if the base has infinite entropy, which in the amenable case is well known and in the case that the acting group contains the free group on two generators was established by Bowen using a different argument.
For a closed cocompact subgroup $Gamma$ of a locally compact group $G$, given a compact abelian subgroup $K$ of $G$ and a homomorphism $rho:hat{K}to G$ satisfying certain conditions, Landstad and Raeburn constructed equivariant noncommutative deforma
tions $C^*(hat{G}/Gamma, rho)$ of the homogeneous space $G/Gamma$, generalizing Rieffels construction of quantum Heisenberg manifolds. We show that when $G$ is a Lie group and $G/Gamma$ is connected, given any norm on the Lie algebra of $G$, the seminorm on $C^*(hat{G}/Gamma, rho)$ induced by the derivation map of the canonical $G$-action defines a compact quantum metric. Furthermore, it is shown that this compact quantum metric space depends on $rho$ continuously, with respect to quantum Gromov-Hausdorff distances.
We show that every infinite-dimensional commutative unital C*-algebra has a Hilbert C*-module admitting no frames. In particular, this shows that Kasparovs stabilization theorem for countably generated Hilbert C*-modules can not be extended to arbitrary Hilbert C*-modules.
We prove a conjecture of Rosenberg about the minimal value for energies of untaries in the two-dimensional noncommutative tori and answer a question of his about lower bounds for energies of unital *-endomorphisms of the two-dimensional noncommutative tori.
