[Paper Review] The maximal coarse Baum-Connes conjecture for spaces which admit a fibred coarse embedding into Hilbert space

Xiaoman Chen, Qin Wang+1 more|arXiv (Cornell University)|Aug 22, 2012

Advanced Operator Algebra Research22 references6 citations

TL;DR

This paper introduces the notion of fibred coarse embedding into Hilbert space, a generalization of Gromov's coarse embedding, and proves the maximal coarse Baum-Connes conjecture for discrete metric spaces with bounded geometry that admit such an embedding. The key result establishes the conjecture for a broad class of spaces, including certain expander graphs, by reducing the problem to twisted algebras at infinity and using homotopy invariance and Bott periodicity in K-theory.

ABSTRACT

We introduce a notion of fibred coarse embedding into Hilbert space for metric spaces, which is a generalization of Gromov's notion of coarse embedding into Hilbert space. It turns out that a large class of expander graphs admit such an embedding. We show that the maximal coarse Baum-Connes conjecture holds for metric spaces with bounded geometry which admit a fibred coarse embedding into Hilbert space.

Motivation & Objective

• To introduce and study the notion of fibred coarse embedding into Hilbert space as a generalization of Gromov's coarse embedding.
• To prove the maximal coarse Baum-Connes conjecture for discrete metric spaces with bounded geometry admitting such an embedding.
• To extend existing results on the coarse Baum-Connes conjecture to include certain expander graphs not coarsely embeddable into Hilbert space.
• To provide a unified framework using twisted Roe algebras and localization algebras at infinity for verifying the conjecture.

Proposed method

• Introduce fibred coarse embedding as a local-to-global condition where large bounded subsets of a space can be coarsely embedded into Hilbert space with uniform distortion at infinity.
• Reduce the maximal coarse Baum-Connes conjecture to verifying isomorphism of K-theory maps for the localization algebra and maximal Roe algebra at infinity of a coarse disjoint union of finite subspaces.
• Define twisted Roe algebras and their localization algebras for sequences of finite metric spaces with fibred coarse embedding into Hilbert space.
• Construct asymptotic morphisms and homotopies between them, particularly involving the Bott-Dirac operator and unitary conjugation, to establish K-theory isomorphisms.
• Use a geometric analogue of Bott periodicity to relate the evaluation map on the localization algebra to the maximal Roe algebra at infinity.
• Apply diagram chasing arguments to conclude that the composition of relevant maps induces the identity on K-theory, proving the conjecture.

Experimental results

Research questions

• RQ1 Can the maximal coarse Baum-Connes conjecture be established for spaces that are not coarsely embeddable into Hilbert space but admit a fibred coarse embedding into Hilbert space?
• RQ2 What structural properties of metric spaces allow the maximal coarse Baum-Connes conjecture to hold despite the failure of classical coarse embeddability?
• RQ3 How can twisted algebras and asymptotic morphisms be used to verify the injectivity and surjectivity of the assembly map in the maximal Roe algebra setting?
• RQ4 To what extent does the fibred coarse embedding condition generalize previous sufficient conditions for the coarse Baum-Connes conjecture?
• RQ5 Can the geometric analogue of Bott periodicity be applied to non-compact, infinite-dimensional metric spaces to stabilize K-theory computations?

Key findings

• The maximal coarse Baum-Connes conjecture holds for any discrete metric space with bounded geometry that admits a fibred coarse embedding into Hilbert space.
• The fibred coarse embedding condition is strictly weaker than Gromov's coarse embeddability into Hilbert space, as it allows certain expander graphs to satisfy the condition.
• The evaluation map from the K-theory of the localization algebra to the K-theory of the maximal Roe algebra at infinity is an isomorphism for coarse disjoint unions of finite metric spaces with fibred coarse embedding.
• The proof relies on constructing a homotopy between asymptotic morphisms involving the Bott-Dirac operator and unitary conjugation, leading to K-theory trivialization.
• The twisted algebra framework allows the reduction of the conjecture to finite subspaces, enabling the use of localization and K-theory techniques.
• The result generalizes prior work by Willett-Yu and Oyono-Oyono-Yu on the maximal coarse Baum-Connes conjecture for spaces of large girth and certain expander graphs.