Yu's Property Research Articles

Quasi-locality and Property A

Let X be a metric space with bounded geometry, p∈{0}∪[1,∞], and let E be a Banach space. The main result of this paper is that either if X has Yu's Property A and p∈(1,∞), or without any condition on X when p∈{0,1,∞}, then quasi-local operators on ℓp(X,E) belong to (the appropriate variant of) the Roe algebra of X. This generalises the existing results of this type by Lange and Rabinovich, Engel, Tikuisis and the first author, and Li, Wang and the second author. As consequences, we obtain that uniform ℓp-Roe algebras (of spaces with Property A) are closed under taking inverses, and another condition characterising Property A, akin to the operator norm localisation for quasi-local operators.

Rigidity of ℓp Roe-type algebras

We investigate the rigidity of the $\ell^p$ analog of Roe-type algebras. In particular, we show that if $p\in[1,\infty)\setminus\{2\}$, then an isometric isomorphism between the $\ell^p$ uniform Roe algebras of two metric spaces with bounded geometry yields a bijective coarse equivalence between the underlying metric spaces, while a stable isometric isomorphism yields a coarse equivalence. We also obtain similar results for other $\ell^p$ Roe-type algebras. In this paper, we do not assume that the metric spaces have Yu's property A or finite decomposition complexity.

On Banach spaces with Kasparov and Yu's Property (H)

A new geometric property of Banach spaces recently introduced by G. Kasparov and G. Yu, called Property (H), has important applications to the strong Novikov conjecture and the coarse Novikov conjecture, yet is far from being well understood. In this paper, we investigate various uniformly continuous maps on unit spheres of Banach spaces and prove that all separable Banach lattices with nontrivial cotype have Property (H).

Almost quasi-isometries and more non-C*-exact groups

AbstractWe study permanence results for almost quasi-isometries, the maps arising from the Gromov construction of finitely generated random groups that contain expanders (and hence that are notC*-exact). We show that the image of a sequence of finite graphs of large girth and controlled vertex degrees under an almost quasi-isometry does not have Guoliang Yu's property A. We use this result to broaden the application of Gromov's techniques to sequences of graphs which are not necessarily expanders. Thus, we obtain more examples of finitely generated non-C*-exact groups.

A metric approach to limit operators

We extend the limit operator machinery of Rabinovich, Roch, and Silbermann from Z^N to (bounded geometry, strongly) discrete metric spaces. We do not assume the presence of any group structure or action on our metric spaces. Using this machinery and recent ideas of Lindner and Seidel, we show that if a metric space X has Yu's property A, then a band-dominated operator on X is Fredholm if and only if all of its limit operators are invertible. We also show that this always fails for metric spaces without property A

An intermediate quasi-isometric invariant between subexponential asymptotic dimension growth and Yu's property A

We present a new quasi-isometric invariant for metric spaces that we name asymptotically large depth. For finitely generated groups we show that this invariant implies Yu's property A and is implied by subexponential asymptotic dimension growth.

AMENABLE ACTIONS, INVARIANT MEANS AND BOUNDED COHOMOLOGY

We show that amenability of an action of a discrete group on a compact space X is equivalent to vanishing of bounded cohomology for a class of Banach G-modules associated to the action, that can be viewed as analogs of continuous bundles of dual modules over the G-space X. In the case when the compact space is a point, our result reduces to a classic theorem of Johnson, characterising amenability of groups. In the case when the compact space is the Stone–Čech compactification of the group, we obtain a cohomological characterisation of exactness, or equivalently, Yu's property A for the group, answering a question of Higson.

Asymptotic dimension, property A, and Lipschitz maps

It is well-known that a paracompact space X is of covering dimension n if and only if any map f from X to a simplicial complex K can be pushed into its n-skeleton. We use the same idea to define dimension in the coarse category. It turns out the analog of maps f from X to K is related to asymptotically Lipschitz maps, the analog of paracompact spaces are spaces related to Yu's Property A, and the dimension coincides with Gromov's asymptotic dimension.

METRIC SPACES WITH SUBEXPONENTIAL ASYMPTOTIC DIMENSION GROWTH

We prove that a metric space with subexponential asymptotic dimension growth has Yu's property A.

Expanders and property A

We give a cohomological characterisation of expander graphs, and use it to give a direct proof that expander graphs do not have Yu's property A.

Property A and affine buildings

Yu's Property A is a non-equivariant generalisation of amenability introduced in his study of the coarse Baum Connes conjecture. In this paper we show that all affine buildings of type A ˜ 2 , B ˜ 2 and G ˜ 2 have Property A. Together with results of Guentner, Higson and Weinberger, this completes a programme to show that all affine building have Property A. In passing we use our technique to obtain a new proof for groups acting on A ˜ n buildings.

Coarsely embeddable metric spaces without Property A

We study Guoliang Yu's Property A and construct metric spaces which do not satisfy Property A but embed coarsely into the Hilbert space.

On Ozawa kernels

We write explicitly Ozawa kernels for group extensions, for discrete metric spaces of finite asymptotic dimension, of large enough Hilbert space compression, and for suitable actions of countable groups on metric spaces. We also obtain an alternative proof of stability results concerning Yu's property A.

RANK DISTRIBUTIONS ON COARSE SPACES AND IDEAL STRUCTURE OF ROE ALGEBRAS

The notions of controlled truncations for operators in the Roe algebras C* (X) of a coarse space (X, ε) with uniformly locally finite coarse structure, and rank distributions on (X, ε) are introduced. It is shown that the controlled propagation operators in an ideal I of C* (X) are exactly the controlled truncations of elements in I. It follows that the lattice of the ideals of C* (X) in which controlled propagation operators are dense is isomorphic to the lattice of all rank distributions on (X, ε). If X is a discrete metric space with Yu's property A, then the ideal structure of the Roe algebra C* (X is completely determined by the rank distributions on X. 2000 Mathematics Subject Classification 46L80, 46L89.

EXACTNESS OF FREE AND AMENABLE GROUPS BY THE CONSTRUCTION OF OZAWA KERNELS

Using properties of their Cayley graphs, specific examples of Ozawa kernels are constructed for both free and amenable groups, thus showing that these groups satisfy Property $O$. It is deduced both that these groups are exact and satisfy Yu's Property $A$.

Hilbert space compression and exactness of discrete groups

We show that the Hilbert space compression of any (unbounded) finite-dimensional CAT(0) cube complex is 1 and deduce that any finitely generated group acting properly, co-compactly on a CAT(0) cube complex is exact, and hence has Yu's Property A. The class of groups covered by this theorem includes free groups, finitely generated Coxeter groups, finitely generated right angled Artin groups, finitely presented groups satisfying the B(4)–T(4) small cancellation condition and all those word-hyperbolic groups satisfying the B(6) condition. Another family of examples is provided by certain canonical surgeries defined by link diagrams.