Ghostly Ideals in Operator Algebras

• Ghostly ideals are defined by relaxing strict support conditions to capture asymptotic invisibility in uniform Roe, Roe, and ℓp algebras.
• They form the upper boundary in ideal lattices, contrasting with geometric ideals and reflecting rank distributions and directional vanishing.
• Extensions to ℓp settings and links to rigidity, K-theory, and Property A illustrate their practical impact in coarse-geometric analysis.

Ghostly ideals are ideals in uniform Roe algebras, Roe algebras, and their $\ell^p$-analogues that formalize asymptotic invisibility in prescribed directions at infinity. They were introduced in contrast to geometric ideals: for a fixed coarse-geometric support datum, the geometric ideal is the smallest compatible ideal, while the ghostly ideal is the largest. In the uniform Roe setting, for an invariant open subset $U\subseteq \beta X$, the defining formula is

$$\tilde I(U):= \{T\in C_u^*(X): \overline{r(\operatorname{supp}_\varepsilon(T))}\subseteq U \text{ for every }\varepsilon>0\},$$

and this is equivalent to vanishing in the $(\beta X\setminus U)$-direction in the sense that $\Phi_\omega(T)=0$ for all $\omega\in \beta X\setminus U$ (Wang et al., 2023). The notion has since been extended to Roe algebras via rank distributions and to $\ell^p$ uniform Roe algebras via coarse-structure ideals, where it plays the role of the upper boundary in the corresponding ideal fibres (Wang et al., 23 Jul 2025, Chung et al., 10 Jun 2026).

1. Definition and basic framework

The ambient objects are coarse-algebraic operator algebras associated to discrete metric spaces of bounded geometry. In the uniform Roe setting, if $T\in B(\ell^2(X))$ is written as a matrix $T=(T(x,y))_{x,y\in X}$, then

$$\operatorname{supp}(T):=\{(x,y)\in X\times X:T(x,y)\neq 0\},$$

$U\subseteq \beta X$0

and the uniform Roe algebra is the closure of the finite-propagation operators. A closed ideal $U\subseteq \beta X$1 is geometric if $U\subseteq \beta X$2 is dense in $U\subseteq \beta X$3. For invariant open $U\subseteq \beta X$4, the associated geometric ideal is

$U\subseteq \beta X$5

where $U\subseteq \beta X$6 is the Skandalis–Tu–Yu coarse groupoid and $U\subseteq \beta X$7 is the corresponding open subgroupoid (Wang et al., 2023).

Ghostly ideals enlarge this support-controlled picture by replacing exact support conditions with $U\subseteq \beta X$8-support conditions. The same paper gives the basic examples

$U\subseteq \beta X$9

so the usual ghost ideal appears as one special ghostly ideal, while the full algebra appears at the opposite extreme (Wang et al., 2023).

A related formulation, used in the rigidity work on geometric ideals, starts from an ideal $$\tilde I(U):= \{T\in C_u^*(X): \overline{r(\operatorname{supp}_\varepsilon(T))}\subseteq U \text{ for every }\varepsilon>0\},$$0 in the metric space and defines

$$\tilde I(U):= \{T\in C_u^*(X): \overline{r(\operatorname{supp}_\varepsilon(T))}\subseteq U \text{ for every }\varepsilon>0\},$$1

This is the same support-at-threshold philosophy, but expressed directly in terms of row-support lying in a space ideal rather than in an invariant open subset of $$\tilde I(U):= \{T\in C_u^*(X): \overline{r(\operatorname{supp}_\varepsilon(T))}\subseteq U \text{ for every }\varepsilon>0\},$$2 (Jiang et al., 2023).

2. Extremal position in the ideal lattice

A central structural result is that ghostly ideals are extremal objects. For a fixed invariant open subset $$\tilde I(U):= \{T\in C_u^*(X): \overline{r(\operatorname{supp}_\varepsilon(T))}\subseteq U \text{ for every }\varepsilon>0\},$$3, define

$$\tilde I(U):= \{T\in C_u^*(X): \overline{r(\operatorname{supp}_\varepsilon(T))}\subseteq U \text{ for every }\varepsilon>0\},$$4

where

$$\tilde I(U):= \{T\in C_u^*(X): \overline{r(\operatorname{supp}_\varepsilon(T))}\subseteq U \text{ for every }\varepsilon>0\},$$5

Then every ideal with associated open set $$\tilde I(U):= \{T\in C_u^*(X): \overline{r(\operatorname{supp}_\varepsilon(T))}\subseteq U \text{ for every }\varepsilon>0\},$$6 satisfies

$$\tilde I(U):= \{T\in C_u^*(X): \overline{r(\operatorname{supp}_\varepsilon(T))}\subseteq U \text{ for every }\varepsilon>0\},$$7

Accordingly, $$\tilde I(U):= \{T\in C_u^*(X): \overline{r(\operatorname{supp}_\varepsilon(T))}\subseteq U \text{ for every }\varepsilon>0\},$$8 is the smallest element of $$\tilde I(U):= \{T\in C_u^*(X): \overline{r(\operatorname{supp}_\varepsilon(T))}\subseteq U \text{ for every }\varepsilon>0\},$$9, and $(\beta X\setminus U)$0 is the largest (Wang et al., 2023).

This order-theoretic role clarifies a common source of confusion. Ghostly ideals are not merely the ideal of all ghost operators. Rather, the ghost ideal $(\beta X\setminus U)$1 is the special case $(\beta X\setminus U)$2, while for general $(\beta X\setminus U)$3 a ghostly ideal records directional vanishing relative to the complement $(\beta X\setminus U)$4. The paper “Ghostly ideals in uniform Roe algebras” explicitly characterizes this by the equivalence

$(\beta X\setminus U)$5

so the defining asymptotic condition is encoded by limit operators rather than by literal support alone (Wang et al., 2023).

The same extremal pattern persists in later generalizations. In Roe algebras, a rank distribution $(\beta X\setminus U)$6 determines a geometric ideal $(\beta X\setminus U)$7 and a ghostly ideal

$(\beta X\setminus U)$8

and every ideal $(\beta X\setminus U)$9 with $\Phi_\omega(T)=0$0 satisfies

$\Phi_\omega(T)=0$1

Here again the geometric ideal is the lower boundary of the fibre and the ghostly ideal is the upper boundary (Wang et al., 23 Jul 2025).

3. Maximal ideals, fibres, and support data

In the uniform Roe algebra of a bounded-geometry metric space, maximal ideals admit a sharp ghostly description. If $\Phi_\omega(T)=0$2 is maximal invariant open, then $\Phi_\omega(T)=0$3 is a maximal ideal; conversely, every maximal ideal is of this form. Equivalently, maximal ideals are precisely

$\Phi_\omega(T)=0$4

for minimal invariant closed subsets $\Phi_\omega(T)=0$5. If $\Phi_\omega(T)=0$6 is a minimal point, then

$\Phi_\omega(T)=0$7

so minimal boundary points parameterize maximal ideals by kernels of limit-operator homomorphisms (Wang et al., 2023).

The Roe-algebra refinement replaces invariant opens by rank distributions and then fibres rank distributions over invariant open subsets. For a rank distribution $\Phi_\omega(T)=0$8,

$\Phi_\omega(T)=0$9

If $\omega\in \beta X\setminus U$0, then

$\omega\in \beta X\setminus U$1

where $\omega\in \beta X\setminus U$2 and $\omega\in \beta X\setminus U$3 are the minimal and maximal rank distributions over $\omega\in \beta X\setminus U$4. This yields a two-step fibring: ideals fibre over rank distributions, and rank distributions fibre over invariant open subsets (Wang et al., 23 Jul 2025).

An $\omega\in \beta X\setminus U$5 version of the same philosophy appears in the 2026 study of $\omega\in \beta X\setminus U$6 uniform Roe algebras. For a coarse-structure ideal $\omega\in \beta X\setminus U$7,

$\omega\in \beta X\setminus U$8

is the largest ideal in the block

$\omega\in \beta X\setminus U$9

while

$\ell^p$0

is the smallest (Chung et al., 10 Jun 2026).

Setting	Geometric ideal	Ghostly ideal
Uniform Roe algebra	$\ell^p$1	$\ell^p$2
Roe algebra with rank distribution $\ell^p$3	$\ell^p$4	$\ell^p$5
$\ell^p$6 uniform Roe algebra	$\ell^p$7	$\ell^p$8

This repeated lower-bound/upper-bound pattern is the main structural signature of ghostly ideals across the subject (Wang et al., 2023, Wang et al., 23 Jul 2025, Chung et al., 10 Jun 2026).

4. Rigidity and coarse-geometric functoriality

Rigidity theory for ghostly ideals is substantially less complete than for geometric ideals. For geometric ideals in uniform Roe algebras, stable isomorphism recovers coarse equivalence of the associated coarse spaces. By contrast, the ghostly setting admits only a partial result. If $\ell^p$9 is a coarse equivalence and $T\in B(\ell^2(X))$0 are ideals in the metric spaces such that

$T\in B(\ell^2(X))$1

then the associated ghostly ideals $T\in B(\ell^2(X))$2 and $T\in B(\ell^2(X))$3 are Morita equivalent (Jiang et al., 2023).

The same paper explicitly leaves open the natural converse and analogue questions. If $T\in B(\ell^2(X))$4 and $T\in B(\ell^2(X))$5 are coarsely equivalent, it is not known in general whether $T\in B(\ell^2(X))$6 and $T\in B(\ell^2(X))$7 must be Morita equivalent. Conversely, if the ghostly ideals are isomorphic or stably isomorphic, it is not known in general whether the associated coarse spaces must be coarsely equivalent (Jiang et al., 2023).

A useful technical reason for this gap is already visible in the ordinary ghost ideal. The rigidity paper notes that the coarse-structure invariant $T\in B(\ell^2(X))$8 does not distinguish the ghost ideal from the compact ideal: $T\in B(\ell^2(X))$9 This shows that support-based invariants which completely classify geometric ideals can collapse genuinely ghost phenomena. A plausible implication is that ghostly ideals retain asymptotic smallness data not visible at the purely geometric level; the paper itself formulates this as an open problem rather than as a theorem (Jiang et al., 2023).

5. Property A, partial Property A, and $T=(T(x,y))_{x,y\in X}$0-theory

The relationship between geometric and ghostly ideals is particularly sharp under amenability-type hypotheses. In the uniform Roe setting, if $T=(T(x,y))_{x,y\in X}$1 coarsely embeds into Hilbert space, then for every invariant open $T=(T(x,y))_{x,y\in X}$2, the inclusion

$T=(T(x,y))_{x,y\in X}$3

induces an isomorphism

$T=(T(x,y))_{x,y\in X}$4

Applied to $T=(T(x,y))_{x,y\in X}$5, this yields

$T=(T(x,y))_{x,y\in X}$6

so the compact ideal and the ghost ideal can differ as ideals while remaining $T=(T(x,y))_{x,y\in X}$7-theoretically indistinguishable (Wang et al., 2023).

The same paper introduces partial Property A toward $T=(T(x,y))_{x,y\in X}$8, defined by amenability of the reduced boundary subgroupoid $T=(T(x,y))_{x,y\in X}$9. For countably generated invariant open $$\operatorname{supp}(T):=\{(x,y)\in X\times X:T(x,y)\neq 0\},$$0, the following are equivalent:

1. $$\operatorname{supp}(T):=\{(x,y)\in X\times X:T(x,y)\neq 0\},$$1 has partial Property A toward $$\operatorname{supp}(T):=\{(x,y)\in X\times X:T(x,y)\neq 0\},$$2;
2. $$\operatorname{supp}(T):=\{(x,y)\in X\times X:T(x,y)\neq 0\},$$3;
3. the ghost ideal $$\operatorname{supp}(T):=\{(x,y)\in X\times X:T(x,y)\neq 0\},$$4 is contained in $$\operatorname{supp}(T):=\{(x,y)\in X\times X:T(x,y)\neq 0\},$$5. This extends the Roe–Willett criterion for $$\operatorname{supp}(T):=\{(x,y)\in X\times X:T(x,y)\neq 0\},$$6 (Wang et al., 2023).

In Roe algebras the same pattern reappears. If $$\operatorname{supp}(T):=\{(x,y)\in X\times X:T(x,y)\neq 0\},$$7 coarsely embeds into Hilbert space, then for every rank distribution $$\operatorname{supp}(T):=\{(x,y)\in X\times X:T(x,y)\neq 0\},$$8,

$$\operatorname{supp}(T):=\{(x,y)\in X\times X:T(x,y)\neq 0\},$$9

Moreover,

$U\subseteq \beta X$00

The paper also gives an expander counterexample showing that outside the coarse-embeddable regime the inclusion need not be a $U\subseteq \beta X$01-isomorphism (Wang et al., 23 Jul 2025).

For coarse embeddings into $U\subseteq \beta X$02-spaces, the $U\subseteq \beta X$03-theoretic comparison was extended further. If $U\subseteq \beta X$04 admits a coarse embedding into an $U\subseteq \beta X$05-space, $U\subseteq \beta X$06, then the canonical inclusion from any geometric ideal to the corresponding ghostly ideal induces a $U\subseteq \beta X$07-theory isomorphism. The same paper deduces relative and maximal coarse Baum–Connes consequences and the property $U\subseteq \beta X$08 (Guo et al., 27 Nov 2025).

6. $U\subseteq \beta X$09 uniform Roe algebras and the groupoid picture

The 2026 theory of $U\subseteq \beta X$10 uniform Roe algebras places ghostly ideals in a broader Banach-algebraic framework. For $U\subseteq \beta X$11, the lattice of geometric ideals in

$U\subseteq \beta X$12

is isomorphic to the lattice of ideals of the coarse structure $U\subseteq \beta X$13. Under the canonical isometric isomorphism

$U\subseteq \beta X$14

geometric ideals correspond precisely to dynamical ideals, while ghostly ideals correspond precisely to restrictive ideals. The same paper gives the limit-operator characterization

$U\subseteq \beta X$15

which is the $U\subseteq \beta X$16 analogue of the vanishing-in-directions description from the Hilbert-space case (Chung et al., 10 Jun 2026).

The role of Property A is also $U\subseteq \beta X$17-dependent. For $U\subseteq \beta X$18, Property A implies that $U\subseteq \beta X$19 admits a multiplier approximate identity with controlled propagation, that all ideals are geometric, and that all ghosts are trivial. For the extreme cases $U\subseteq \beta X$20, these properties hold for every uniformly locally finite coarse space without assuming Property A (Chung et al., 10 Jun 2026).

This establishes the current scope of the subject. Ghostly ideals now appear in three tightly related but distinct guises: as directional-vanishing enlargements of geometric ideals in uniform Roe algebras, as upper fibre boundaries in Roe algebras indexed by rank distributions, and as restrictive ideals in $U\subseteq \beta X$21 groupoid operator algebras. What remains unresolved is not their existence or basic structure, but a full rigidity theory comparable to the one already available for geometric ideals (Jiang et al., 2023).