[Paper Review] Quasi-local Algebras and Asymptotic Expanders

Kang Li, Piotr W. Nowak+2 more|arXiv (Cornell University)|Aug 21, 2019

Advanced Operator Algebra Research37 references4 citations

TL;DR

This paper introduces asymptotic expanders—sequences of finite metric spaces that generalize expanders—and establishes their role in distinguishing uniform Roe algebras from uniform quasi-local algebras. It proves that the averaging projection over a coarse disjoint union belongs to the uniform quasi-local algebra if and only if the sequence is asymptotic expanders, and shows that such algebras are nuclear precisely when the underlying space has Property A.

ABSTRACT

In this paper, we study the relation between the uniform Roe algebra and the uniform quasi-local algebra associated to a metric space of bounded geometry. In the process, we introduce a weakening of the notion of expanders, called asymptotic expanders. We show that being a sequence of asymptotic expanders is a coarse property under certain connectedness condition, and it implies non-uniformly local amenability. Moreover, we also analyse some $C^*$-algebraic properties of uniform quasi-local algebras. In particular, we show that a uniform quasi-local algebra is nuclear if and only if the underlying metric space has Property A.

Motivation & Objective

• To understand the structural difference between the uniform Roe algebra and the uniform quasi-local algebra on metric spaces of bounded geometry.
• To identify obstructions to the equality $ C^{*}_{u}(X) = C^{*}_{uq}(X) $, particularly via the averaging projection $ P_X $.
• To introduce and study the new class of asymptotic expanders as a generalization of expanders, capturing coarse geometric rigidity without full expansion.
• To analyze $ C^* $-algebraic properties of uniform quasi-local algebras, especially nuclearity and Cartan subalgebra structure.
• To investigate the interplay between coarse geometry, quasi-locality, and embeddability into Hilbert spaces.

Proposed method

• Define asymptotic expanders via a uniform isoperimetric condition: for any $ \alpha > 0 $, there exist $ c \in (0,1) $, $ R > 0 $, such that $ |\partial_R A| > c|A| $ for all $ A \subseteq X_n $ with $ \alpha|X_n| \leq |A| \leq |X_n|/2 $.
• Prove that the averaging projection $ P_X $ on a coarse disjoint union $ X = \bigsqcup X_n $ is quasi-local if and only if $ \{X_n\} $ is a sequence of asymptotic expanders.
• Use the normalizer of $ \ell^\infty(X) $ in $ \mathfrak{B}(\ell^2(X)) $ to characterize elements in $ C^{*}_{uq}(X) $, showing it coincides with the normalizer in $ C^{*}_{u}(X) $.
• Establish that $ C^{*}_{uq}(X) $ is nuclear if and only if $ X $ has Property A, using the characterization of nuclearity via asymptotic expanders and quasi-locality.
• Analyze the Cartan pair structure of $ \ell^\infty(X) \subseteq C^{*}_{uq}(X) $, proving it is a Cartan inclusion via normalizer generation.
• Apply results from Finn-Sell and others to show that if $ X $ is coarsely embeddable into a Hilbert space, then $ P_X \notin C^{*}_{u}(X) $, hence $ C^{*}_{u}(X) \subsetneq C^{*}_{uq}(X) $ if $ X $ is a coarse union of asymptotic expanders with $ P_X \in C^{*}_{uq}(X) $.

Experimental results

Research questions

• RQ1 When does the averaging projection $ P_X $ on a coarse disjoint union $ X = \bigsqcup X_n $ belong to the uniform quasi-local algebra $ C^{*}_{uq}(X) $?
• RQ2 Can a sequence of asymptotic expanders be coarsely embedded into a Hilbert space, and what does this imply for the inclusion $ C^{*}_{u}(X) \subseteq C^{*}_{uq}(X) $?
• RQ3 Is the uniform quasi-local algebra $ C^{*}_{uq}(X) $ nuclear if and only if the underlying space $ X $ has Property A?
• RQ4 Does the Cartan pair structure $ \ell^\infty(X) \subseteq C^{*}_{uq}(X) $ hold, and what does it imply about the algebraic generation of $ C^{*}_{uq}(X) $?
• RQ5 Can a space $ X $ with $ C^{*}_{u}(X) \subsetneq C^{*}_{uq}(X) $ be coarsely embeddable into a Hilbert space, and what would this imply for the coarse Baum-Connes conjecture?

Key findings

• The averaging projection $ P_X $ on a coarse disjoint union $ X = \bigsqcup X_n $ is quasi-local if and only if $ \{X_n\} $ is a sequence of asymptotic expanders.
• Asymptotic expanders are strictly more general than expanders, as shown by a construction of a sequence satisfying the asymptotic condition but not the standard expander condition.
• Being a sequence of asymptotic expanders is a coarse invariant under coarse equivalences when the graphs are connected and have bounded valency.
• A sequence of asymptotic expanders is incompatible with uniformly local amenability, implying a coarse geometric obstruction to amenability.
• The uniform quasi-local algebra $ C^{*}_{uq}(X) $ is nuclear if and only if the underlying metric space $ X $ has Property A.
• If $ X $ is a coarse disjoint union of asymptotic expanders and $ C^{*}_{u}(X) = C^{*}_{uq}(X) $, then $ X $ cannot be coarsely embedded into any Hilbert space, as $ P_X \in C^{*}_{u}(X) $ would contradict the non-existence of non-compact ghost projections in such Roe algebras.