Stable Coarse Algebras in Coarse Geometry

• Stable coarse algebras are C*-subalgebras defined over discrete metric spaces that are closed under bounded B(H)-multiplication and compact stabilization.
• They serve as coefficient systems for twisted Roe algebras, ensuring invariance under coarse transport and enabling coarse Baum–Connes conjecture formulations.
• The framework allows for robust decomposition arguments and permanence results, playing a key role in relative hyperbolicity and coarse index theory.

Stable coarse algebras are $C^*$-subalgebras of $\ell^\infty(X,\mathcal A\otimes \mathcal K)$ attached to a discrete metric space $X$, introduced as a refinement of the coarse-algebra framework for twisted Roe algebras. Their defining feature is not merely invariance under coarse transport by partial translations, but additional closure under bounded $B(\mathcal H)$-multiplication and stabilization by compact operators. In the formulation of Deng, Guo, and Wang, these extra axioms are introduced precisely to guarantee coarse invariance of twisted Roe algebras, independence of the choice of net and Borel cover, and permanence under subspaces and unions; they then serve as the coefficient objects for a twisted coarse Baum–Connes conjecture and for a relatively hyperbolic permanence theorem (Deng et al., 8 Sep 2025).

1. Definition and basic structure

The notion is defined for a discrete metric space $X$ and a $C^*$-algebra $\mathcal A$. One fixes a separable infinite-dimensional Hilbert space $\mathcal H$, the compact operators $\mathcal K=\mathcal K(\mathcal H)$, and a fixed $*$-isomorphism

$\ell^\infty(X,\mathcal A\otimes \mathcal K)$0

extended to

$\ell^\infty(X,\mathcal A\otimes \mathcal K)$1

A $\ell^\infty(X,\mathcal A\otimes \mathcal K)$2-subalgebra

$\ell^\infty(X,\mathcal A\otimes \mathcal K)$3

is a coarse $\ell^\infty(X,\mathcal A\otimes \mathcal K)$4-algebra if it satisfies two conditions. First, it has fiberwise fullness: $\ell^\infty(X,\mathcal A\otimes \mathcal K)$5 Second, it is invariant under partial translations: if $\ell^\infty(X,\mathcal A\otimes \mathcal K)$6 is a bijection with

$\ell^\infty(X,\mathcal A\otimes \mathcal K)$7

and

$\ell^\infty(X,\mathcal A\otimes \mathcal K)$8

then $\ell^\infty(X,\mathcal A\otimes \mathcal K)$9 implies $X$0 (Deng et al., 8 Sep 2025).

A coarse $X$1-algebra is called stable if, in addition, it satisfies two further axioms. The first is $X$2-multiplicative stability: for bounded sequences $X$3 and $X$4, one defines

$X$5

and requires that

$X$6

The second is stability under adding a compact tensor factor: for every $X$7 and $X$8, the function

$X$9

must again belong to $B(\mathcal H)$0. The authors state explicitly that conditions (3) and (4) are used to guarantee the coarse invariance of twisted Roe algebras (Deng et al., 8 Sep 2025).

The basic examples are the maximal coefficient system $B(\mathcal H)$1, the vanishing-at-infinity system $B(\mathcal H)$2, and the closure of the neighborhood-supported algebra associated to a subspace $B(\mathcal H)$3,

$B(\mathcal H)$4

where $B(\mathcal H)$5 (Deng et al., 8 Sep 2025).

2. Twisted Roe algebras with stable coarse coefficients

Stable coarse algebras are designed to enter the Roe-algebra construction as coefficient systems. The ambient space for the operator algebra is a metric space $B(\mathcal H)$6 with bounded geometry together with a chosen net $B(\mathcal H)$7, meaning that there exist $B(\mathcal H)$8 such that distinct points of $B(\mathcal H)$9 are at distance at least $X$0 and $X$1. One then chooses a Borel decomposition $X$2 such that each $X$3 contains a nonempty open subset, the $X$4 are pairwise disjoint, each $X$5 belongs to $X$6, and $X$7 (Deng et al., 8 Sep 2025).

With a countable dense subset $X$8 and the standard Hilbert $X$9-module

$C^*$0

the relevant module is $C^*$1. Adjointable $C^*$2-linear operators on this module are matrices

$C^*$3

Propagation is defined by

$C^*$4

and local compactness is defined by compactness of every bounded truncation. The ordinary Roe algebra with coefficients $C^*$5 is then the $C^*$6-subalgebra generated by locally compact finite-propagation operators,

$C^*$7

The twisted construction replaces constant coefficients by the stable coarse algebra $C^*$8 (Deng et al., 8 Sep 2025).

The twisting condition is encoded blockwise. Choosing unitaries

$C^*$9

and a partial translation $\mathcal A$0 on $\mathcal A$1, one defines

$\mathcal A$2

The algebraic twisted Roe algebra $\mathcal A$3 consists of locally compact finite-propagation operators $\mathcal A$4 such that

$\mathcal A$5

and the completed twisted Roe algebra is its operator-norm closure

$\mathcal A$6

A decisive point is that when $\mathcal A$7 is stable, the condition $\mathcal A$8 is independent of the choice of the unitaries $\mathcal A$9, because changing the $\mathcal H$0 multiplies by bounded $\mathcal H$1-valued sequences, exactly the situation covered by $\mathcal H$2-multiplicative stability (Deng et al., 8 Sep 2025).

This coefficient-theoretic use of stability sits inside the broader Roe-algebraic setting developed for coarse spaces via coarse geometric modules. Martínez and Vigolo construct Roe-like algebras for general coarse spaces, define controlled propagation, local compactness, and quasi-locality in that generality, and explicitly remark that one may similarly define stable Roe algebras, although their technical development is restricted to ordinary Roe algebras, controlled-propagation algebras, and quasi-local algebras (Martínez et al., 2023).

3. Coarse invariance, pullback, and permanence

A central reason for introducing stable coarse algebras is that they can be transported along coarse equivalences. If $\mathcal H$3 is a coarse equivalence and $\mathcal H$4 is a stable coarse $\mathcal H$5-algebra, the pullback coefficient system on $\mathcal H$6 is defined by requiring that for every subset $\mathcal H$7 of representatives of the fibers of $\mathcal H$8, the associated function $\mathcal H$9 on $\mathcal K=\mathcal K(\mathcal H)$0 lies in $\mathcal K=\mathcal K(\mathcal H)$1: $\mathcal K=\mathcal K(\mathcal H)$2 The paper proves that this is again a stable coarse $\mathcal K=\mathcal K(\mathcal H)$3-algebra, that close coarse equivalences induce the same pulled-back coefficient system, and that pullback is compatible with composition (Deng et al., 8 Sep 2025).

This pullback mechanism is the basis of several invariance theorems. If $\mathcal K=\mathcal K(\mathcal H)$4 and $\mathcal K=\mathcal K(\mathcal H)$5 are two nets in the same metric space $\mathcal K=\mathcal K(\mathcal H)$6, then after transporting coefficients along a coarse equivalence close to the identity, one has

$\mathcal K=\mathcal K(\mathcal H)$7

inside the same operator algebra. More generally, if $\mathcal K=\mathcal K(\mathcal H)$8 is a coarse equivalence between bounded-geometry metric spaces, then for any stable coarse $\mathcal K=\mathcal K(\mathcal H)$9-algebra there exists a stable coarse $*$0-algebra such that

$*$1

The same framework yields permanence of the twisted coarse Baum–Connes conjecture under coarse equivalence, subspaces, and unions (Deng et al., 8 Sep 2025).

The subspace theorem is especially characteristic. If $*$2, one uses coarse equivalences $*$3 fixing $*$4, transports $*$5 to each thickening $*$6, and forms

$*$7

The resulting ambient twisted Roe algebra satisfies

$*$8

This identifies the Roe algebra of a subspace as a twisted Roe algebra on the ambient space, a feature used repeatedly in later decomposition arguments (Deng et al., 8 Sep 2025).

For families of subspaces, the paper builds stable coarse coefficient systems with coefficient algebra $*$9, thereby encoding a disjoint union $\ell^\infty(X,\mathcal A\otimes \mathcal K)$00 by a single stable coefficient algebra on one space. This is then used to pass from single-space statements to uniform statements about families (Deng et al., 8 Sep 2025).

4. The twisted coarse Baum–Connes conjecture

The stable-coefficient formalism is used to formulate a twisted coarse Baum–Connes conjecture. For a bounded-geometry discrete metric space $\ell^\infty(X,\mathcal A\otimes \mathcal K)$01, one considers the Rips complexes $\ell^\infty(X,\mathcal A\otimes \mathcal K)$02. For each $\ell^\infty(X,\mathcal A\otimes \mathcal K)$03, the twisted localization algebra

$\ell^\infty(X,\mathcal A\otimes \mathcal K)$04

is defined as the completion of uniformly bounded, uniformly continuous functions

$\ell^\infty(X,\mathcal A\otimes \mathcal K)$05

such that

$\ell^\infty(X,\mathcal A\otimes \mathcal K)$06

Evaluation at $\ell^\infty(X,\mathcal A\otimes \mathcal K)$07 gives a $\ell^\infty(X,\mathcal A\otimes \mathcal K)$08-homomorphism

$\ell^\infty(X,\mathcal A\otimes \mathcal K)$09

and therefore a map on $\ell^\infty(X,\mathcal A\otimes \mathcal K)$10-theory

$\ell^\infty(X,\mathcal A\otimes \mathcal K)$11

Passing to the direct limit over $\ell^\infty(X,\mathcal A\otimes \mathcal K)$12 yields the twisted coarse assembly map

$\ell^\infty(X,\mathcal A\otimes \mathcal K)$13

The conjecture asserts that this map is an isomorphism for every stable coarse algebra $\ell^\infty(X,\mathcal A\otimes \mathcal K)$14 (Deng et al., 8 Sep 2025).

The obstruction algebra is the kernel of evaluation,

$\ell^\infty(X,\mathcal A\otimes \mathcal K)$15

and the six-term exact sequence gives the criterion

$\ell^\infty(X,\mathcal A\otimes \mathcal K)$16

This is the operative form used in the paper’s Mayer–Vietoris and decomposition arguments (Deng et al., 8 Sep 2025).

The framework contains the usual coarse Baum–Connes setting as a special case. For the maximal stable coarse algebra $\ell^\infty(X,\mathcal A\otimes \mathcal K)$17, one recovers the ordinary Roe algebra with coefficients: $\ell^\infty(X,\mathcal A\otimes \mathcal K)$18 At the opposite extreme, for $\ell^\infty(X,\mathcal A\otimes \mathcal K)$19, the twisted Roe algebra collapses to

$\ell^\infty(X,\mathcal A\otimes \mathcal K)$20

The stable coarse algebra formalism therefore interpolates between maximal coarse coefficients and highly localized coefficient systems (Deng et al., 8 Sep 2025).

5. Relatively hyperbolic groups and decomposition methods

The main application in the introducing paper concerns finitely generated groups $\ell^\infty(X,\mathcal A\otimes \mathcal K)$21 hyperbolic relative to a finite family of subgroups $\ell^\infty(X,\mathcal A\otimes \mathcal K)$22. The principal theorem states that

$\ell^\infty(X,\mathcal A\otimes \mathcal K)$23

if and only if each subgroup $\ell^\infty(X,\mathcal A\otimes \mathcal K)$24 does (Deng et al., 8 Sep 2025).

The proof uses the ordinary word metric $\ell^\infty(X,\mathcal A\otimes \mathcal K)$25, the relative metric $\ell^\infty(X,\mathcal A\otimes \mathcal K)$26, and the relative balls

$\ell^\infty(X,\mathcal A\otimes \mathcal K)$27

A key decomposition is

$\ell^\infty(X,\mathcal A\otimes \mathcal K)$28

Writing

$\ell^\infty(X,\mathcal A\otimes \mathcal K)$29

one applies the Osin/Dadarlat–Guentner separation lemma: for every $\ell^\infty(X,\mathcal A\otimes \mathcal K)$30, there exists $\ell^\infty(X,\mathcal A\otimes \mathcal K)$31 such that if

$\ell^\infty(X,\mathcal A\otimes \mathcal K)$32

then

$\ell^\infty(X,\mathcal A\otimes \mathcal K)$33

and the pieces $\ell^\infty(X,\mathcal A\otimes \mathcal K)$34 are $\ell^\infty(X,\mathcal A\otimes \mathcal K)$35-separated. This is the geometric input for the decomposition-complexity induction (Deng et al., 8 Sep 2025).

Stable coarse algebras are indispensable in this argument for four reasons spelled out by the authors. They allow one to identify Roe algebras of subspaces as twisted Roe algebras on ambient spaces; to replace families and disjoint unions by a single space with a larger stable coefficient algebra; to apply controlled Mayer–Vietoris arguments uniformly across families; and to retain invariance under coarse equivalence and under thickening $\ell^\infty(X,\mathcal A\otimes \mathcal K)$36. The argument is organized via a class $\ell^\infty(X,\mathcal A\otimes \mathcal K)$37 of metric families satisfying the twisted conjecture uniformly with respect to any stable coarse algebras, and finite decomposition complexity of $\ell^\infty(X,\mathcal A\otimes \mathcal K)$38 completes the induction (Deng et al., 8 Sep 2025).

This application clarifies the intended role of stable coarse algebras. They are not merely alternative coefficient algebras; they are the mechanism by which relative, decomposition-based, and family-based arguments become compatible with coarse index theory.

6. Relation to Roe-like algebras and other meanings of “stable”

The term “stable” in stable coarse algebra is coefficient-theoretic. It refers to closure of $\ell^\infty(X,\mathcal A\otimes \mathcal K)$39 under bounded $\ell^\infty(X,\mathcal A\otimes \mathcal K)$40-multiplication and compact stabilization, not to the tensorial absorption property $\ell^\infty(X,\mathcal A\otimes \mathcal K)$41. This distinction is important because Roe-theoretic literature uses “stable” in several different senses.

In the Roe-like setting of Braga, Chung, Vignati, and Willett, “stable” is used in the standard $\ell^\infty(X,\mathcal A\otimes \mathcal K)$42-algebraic sense of stable isomorphism. For countably generated coarse spaces, under the hypotheses of their stable rigidity theorem, an isomorphism

$\ell^\infty(X,\mathcal A\otimes \mathcal K)$43

implies that $\ell^\infty(X,\mathcal A\otimes \mathcal K)$44 and $\ell^\infty(X,\mathcal A\otimes \mathcal K)$45 are coarsely equivalent. Their memoir treats Roe algebras, controlled-propagation algebras, quasi-local algebras, and the stabilized uniform Roe algebra in a unified way, and shows that stabilization by compact operators does not destroy the coarse geometric information encoded by these Roe-like algebras (Martínez et al., 2024).

Martínez and Vigolo, in turn, formulate Roe algebras of general coarse spaces by means of coarse geometric modules. They explicitly note that one may similarly define other Roe-like $\ell^\infty(X,\mathcal A\otimes \mathcal K)$46-algebras of operators, such as stable Roe algebras, but restrict their technical development to the Roe algebra, the algebra of controlled propagation operators, and the quasi-local algebra. Their module-independence results are stronger in a different direction: once a cardinal rank $\ell^\infty(X,\mathcal A\otimes \mathcal K)$47 and ampleness class are fixed, different discrete $\ell^\infty(X,\mathcal A\otimes \mathcal K)$48-ample rank-$\ell^\infty(X,\mathcal A\otimes \mathcal K)$49 modules produce actually $\ell^\infty(X,\mathcal A\otimes \mathcal K)$50-isomorphic Roe-like algebras, not merely stably isomorphic ones (Martínez et al., 2023).

A different ambiguity arises from Banach-algebra terminology. In “Stable normed algebras,” Ansari Piri and Nouri call a normed algebra stable when its norm is recovered from left multiplication,

$\ell^\infty(X,\mathcal A\otimes \mathcal K)$51

equivalently when

$\ell^\infty(X,\mathcal A\otimes \mathcal K)$52

That notion is unrelated to stable coarse algebras in the Roe-theoretic sense, even though every $\ell^\infty(X,\mathcal A\otimes \mathcal K)$53-algebra satisfies it (Piri et al., 2015).

Taken together, these distinctions show that stable coarse algebras occupy a specific niche. They are coarse-geometric coefficient systems for twisted Roe and localization algebras, engineered so that coarse transport, matrix amplification, and compact stabilization are all internal operations. This suggests a precise conceptual role: they provide the coefficient language in which coarse invariance, subspace functoriality, and decomposition arguments can be formulated simultaneously within coarse index theory (Deng et al., 8 Sep 2025).