[Paper Review] Coarse Baum-Connes conjecture and rigidity for Roe algebras

Bruno de Mendonça Braga, Yeong Chyuan Chung+1 more|arXiv (Cornell University)|Jul 24, 2019

Advanced Operator Algebra Research28 references4 citations

TL;DR

This paper establishes a connection between the coarse Baum-Connes conjecture and rigidity of Roe algebras, proving that if two uniformly locally finite metric spaces have $*$-isomorphic Roe algebras and one satisfies the coarse Baum-Connes conjecture with coefficients, then the spaces are coarsely equivalent. The key contribution is extending rigidity results beyond spaces coarsely embeddable in Hilbert space to a broader class defined by ghost projection properties.

ABSTRACT

In this paper, we connect the rigidity problem and the coarse Baum-Connes conjecture for Roe algebras. In particular, we show that if $X$ and $Y$ are two uniformly locally finite metric spaces such that their Roe algebras are $*$-isomorphic, then $X$ and $Y$ are coarsely equivalent provided either $X$ or $Y$ satisfies the coarse Baum-Connes conjecture with coefficients. It is well-known that coarse embeddability into a Hilbert space implies the coarse Baum-Connes conjecture with coefficients. On the other hand, we provide a new example of a finitely generated group satisfying the coarse Baum-Connes conjecture with coefficients but which does not coarsely embed into a Hilbert space.

Motivation & Objective

• To address the rigidity problem for Roe algebras by connecting it to the coarse Baum-Connes conjecture with coefficients.
• To extend known rigidity results beyond spaces coarsely embeddable in Hilbert space to a strictly larger class of metric spaces.
• To characterize a new class of metric spaces where all sparse subspaces yield only compact ghost projections in their Roe algebras.
• To provide a finitely generated group satisfying the coarse Baum-Connes conjecture with coefficients but not coarsely embeddable in Hilbert space, answering an open question.
• To establish uniform Roe bijective rigidity under the coarse Baum-Connes conjecture with coefficients for non-amenable groups.

Proposed method

• Introduce and analyze a geometric condition: all sparse subspaces of a metric space yield only compact ghost projections in their Roe algebras.
• Use the coarse Baum-Connes conjecture with coefficients to rule out noncompact ghost projections in Roe algebras.
• Apply the boundary coarse Baum-Connes assembly map and its relation to the standard Baum-Connes map via groupoid crossed products.
• Leverage the fact that coarse embeddability into Hilbert space implies the geometric condition, but show the condition is strictly weaker.
• Use $*$-isomorphisms between Roe algebras and stable/uniform Roe algebras to deduce coarse equivalence under the conjecture assumption.
• Construct a finitely generated group that satisfies the coarse Baum-Connes conjecture with coefficients but does not coarsely embed into Hilbert space using the geometric condition.

Experimental results

Research questions

• RQ1 Does the coarse Baum-Connes conjecture with coefficients imply that $*$-isomorphic Roe algebras correspond to coarsely equivalent metric spaces?
• RQ2 Is the geometric condition on ghost projections in sparse subspaces strictly weaker than coarse embeddability into Hilbert space?
• RQ3 Can a finitely generated group satisfy the coarse Baum-Connes conjecture with coefficients without coarsely embedding into a Hilbert space?
• RQ4 Under what conditions is a metric space Roe rigid or uniformly Roe rigid based on algebraic isomorphisms of its Roe algebras?
• RQ5 Is uniform Roe bijective rigidity implied by the coarse Baum-Connes conjecture with coefficients for non-amenable groups?

Key findings

• If $X$ and $Y$ are uniformly locally finite metric spaces with $*$-isomorphic Roe algebras and one satisfies the coarse Baum-Connes conjecture with coefficients, then $X$ and $Y$ are coarsely equivalent.
• The geometric condition that all sparse subspaces yield only compact ghost projections is strictly weaker than coarse embeddability into Hilbert space.
• A new example of a finitely generated group is constructed that satisfies the coarse Baum-Connes conjecture with coefficients but does not coarsely embed into a Hilbert space.
• The condition that all sparse subspaces yield only compact ghost projections implies uniform Roe rigidity for uniformly locally finite metric spaces.
• For non-amenable metric spaces satisfying the coarse Baum-Connes conjecture with coefficients, uniform Roe bijective rigidity holds.
• The coarse Baum-Connes conjecture with coefficients implies the absence of noncompact ghost projections in the Roe algebra, which is key to the rigidity results.