Groupoid exactness and the weak containment problem

This paper investigates analogues of exactness for locally compact groupoids, proving that six natural definitions of exactness are equivalent for the class of inner amenable étale groupoids and highlighting the crucial role of exactness in understanding the relationship between groupoid amenability and the coincidence of full and reduced C*-algebras.

The Gist

Imagine you are an architect trying to understand the stability of a massive, complex city. This city isn't made of bricks and mortar, but of mathematical structures called groupoids.

In the world of mathematics, a "groupoid" is like a city map where you can travel between different points (units) using specific routes (arrows). Sometimes, you can go from Point A to Point B, but not necessarily back. Sometimes, the roads loop around. It's a generalization of a "group" (like a set of rotations or symmetries) that allows for more complex, fragmented connections.

This paper, written by Claire Anantharaman-Delarocche, is a deep dive into two major questions about these mathematical cities:

1. Exactness: Is the city structurally sound? Can we break it down into smaller, manageable pieces without the whole thing collapsing?
2. Weak Containment: Does the city have a "perfect" version? In math, every city has a "Full" version (everything that could happen) and a "Reduced" version (only what does happen). The question is: Are these two versions actually the same city?

Here is the breakdown of the paper's journey, explained with everyday analogies.

1. The Problem: Two Versions of the City

Imagine you are designing a theme park.

• The Full Park ($C^*(G)$): This is the blueprint. It includes every possible ride, every theoretical path, and every "what if" scenario. It's the most complete, idealized version.
• The Reduced Park ($C^*_r(G)$): This is the park as it actually operates. It only includes the rides that are built and the paths people actually walk.

For a long time, mathematicians wondered: If the "Reduced" park behaves exactly like the "Full" park (meaning they are mathematically identical), does that mean the park is "Amenable"?

"Amenable" is a fancy math word that roughly means "well-behaved" or "peaceful." A peaceful park is one where you can easily predict the flow of traffic, and there are no chaotic, unpredictable loops.

For simple groups (like a single, solid block of symmetry), the answer was known: Yes, if the parks are identical, the group is peaceful. But for these complex "groupoid" cities, the answer was a mystery.

2. The New Tool: "Amenability at Infinity"

To solve this, the author introduces a new concept called "Amenability at Infinity."

Think of a city that gets more and more chaotic as you travel further out into the suburbs.

• Standard Amenability: The whole city is peaceful, from the center to the edge.
• Amenability at Infinity: The center might be chaotic, but if you look far enough out (at "infinity"), the suburbs are perfectly peaceful and organized.

The paper argues that if a groupoid is "peaceful at infinity," it unlocks a secret door. It turns out that for a specific type of groupoid (called étale, which are like cities with discrete, step-by-step roads), being peaceful at infinity is the key that makes all the different definitions of "structural soundness" (Exactness) line up perfectly.

3. The "Inner" Secret: Inner Amenability

The author discovers that this magic only works if the city has a hidden property called "Inner Amenability."

The Analogy: Imagine a city where every neighborhood has a "ghost" version of itself that mirrors the main streets. If you can find a way to match every street in the main city with its ghost twin without getting lost, the city is "Inner Amenable."

• For simple groups, this is always true.
• For complex groupoids, we don't know if all of them have this property. The author admits, "We don't know if every étale groupoid is inner amenable yet." It's an open mystery.

However, if a groupoid does have this "Inner" property, the paper proves a massive theorem: Six different ways of measuring the city's stability are actually the same thing.

These six ways include:

1. Is it peaceful at infinity?
2. Is the "Uniform Algebra" (a specific mathematical tool used to measure the city) "nuclear" (super flexible and smooth)?
3. Is the "Reduced C*-algebra" (the operating park) "exact" (structurally sound)?
4. Does it satisfy the "Weak Containment Property" (Full = Reduced)?

The Big Reveal: If the city is "Inner Amenable" and "Peaceful at Infinity," then all six conditions are true at once. If one is true, they all are.

4. The "Monster" Examples

The paper also discusses "HLS-groupoids" (named after Higson, Lafforgue, and Skandalis). These are like "Monster Cities."

• They are built from a sequence of smaller, perfect groups.
• Individually, the pieces are perfect.
• But when you stitch them together, the whole city becomes chaotic and "non-amenable" (unpredictable).

The paper shows that for these Monster Cities, the "Full" and "Reduced" parks are identical (they have the Weak Containment Property), BUT the city is NOT peaceful (not amenable).

• Why? Because they lack the "Inner Amenability" (the ghost twins don't match up).
• This solves a long-standing puzzle: You can have a city where the blueprint and the reality match, but the city is still chaotic. The missing ingredient is that specific "Inner" structure.

5. The Conclusion: What We Know and What We Don't

The paper is a massive step forward, but it leaves a few doors open:

• We know: If a groupoid is "Inner Amenable" and "Peaceful at Infinity," then the "Full" and "Reduced" versions are the same if and only if the city is peaceful.
• We don't know: Are all étale groupoids "Inner Amenable"? If they are, then the rules are even simpler. If not, we have to be very careful about which cities we are studying.

Summary for the Layperson

Think of this paper as a guidebook for a new type of city.

• The Goal: To figure out when a complex, chaotic city is actually "well-behaved."
• The Discovery: The author found a special "Peaceful Zone" (Amenability at Infinity). If a city has this zone, and it has a specific "Ghost Mirror" property (Inner Amenability), then the city is structurally perfect in every possible way.
• The Twist: There are "Monster Cities" that look perfect on paper (Full = Reduced) but are actually chaotic inside. They are chaotic because they lack the "Ghost Mirror" property.

This work helps mathematicians understand the hidden architecture of complex systems, showing that sometimes, to know if a system is stable, you have to look at its "infinity" and its "inner self."

Technical Summary

Here is a detailed technical summary of the monograph "Groupoids Exactness and the Weak Containment Problem" by Claire Anantharaman-Delarocche.

1. Problem Statement

The monograph addresses two fundamental problems concerning locally compact groupoids, extending well-known results from the theory of discrete and locally compact groups:

1. The Equivalence of Exactness Notions: For discrete groups, several definitions of "exactness" (e.g., amenability at infinity, nuclearity of the uniform Roe algebra, exactness of the reduced $C^*$-algebra, and Kirchberg-Wassermann exactness) are known to be equivalent. The author investigates whether these equivalences hold for groupoids, specifically étale groupoids (the groupoid analogue of discrete groups).
2. The Weak Containment Problem (WCP): Does the equality of the full and reduced $C^*$-algebras of a groupoid ($C^*(G) = C^*_r(G)$) imply that the groupoid is amenable? While true for locally compact groups (Hulanicki's theorem), it was an open problem for groupoids. Recent counterexamples (HLS-groupoids) showed that WCP does not imply amenability in general. The paper seeks to identify the specific conditions (related to exactness) under which WCP does imply amenability.

2. Methodology and Framework

The author employs a blend of operator algebraic techniques, topological dynamics, and groupoid theory:

• Fibrewise Compactifications: A central technical tool is the generalization of the Stone-Čech compactification to the groupoid setting. The author constructs the Stone-Čech fibrewise compactification $(\beta_r G, r_\beta)$ of the range map $r: G \to G^{(0)}$. This allows the definition of actions on compact spaces fibered over the unit space.
• Inner Amenability: The author introduces the notion of inner amenability for groupoids. This property ensures the existence of positive definite kernels on the product groupoid $G \times G$ that behave like the characteristic function of the diagonal (crucial for discrete groups). This concept is vital for bridging the gap between $C^*$-exactness and amenability at infinity.
• Uniform $C^*$-Algebras: The paper defines the uniform $C^*$-algebra $C^*_u(G)$ (analogous to the uniform Roe algebra) and establishes its isomorphism with the reduced $C^*$-algebra of the semi-direct product $\beta_r G \rtimes G$.
• Generalized Morphisms: The study utilizes generalized morphisms (Hilsum-Skandalis morphisms) and the concept of equivalence of groupoids to transfer properties between different groupoids.

3. Key Contributions and Definitions

A. New Notions of Exactness and Amenability

The paper refines and introduces several concepts:

• Strong Amenability at Infinity: A groupoid $G$ is strongly amenable at infinity if it admits an amenable action on a fibrewise compact space $(Y, p)$ where $p$ admits a continuous section. This is stronger than standard amenability at infinity.
• Inner Exactness: A groupoid is inner exact if the sequence of reduced $C^*$-algebras associated with any closed invariant subset of the unit space is exact. This is a weaker condition than Kirchberg-Wassermann (KW) exactness but holds for all locally compact groups.
• Inner Amenability: Defined via the existence of specific positive definite functions on $G \times G$. The author proves that all discrete groups are inner amenable, but the status for general étale groupoids remains an open question.

B. The Main Equivalence Theorem

The central result (Theorem 10.8) establishes that for second countable, inner amenable, étale groupoids, the following six conditions are equivalent:

1. Strong amenability at infinity.
2. Amenability at infinity.
3. Nuclearity of the uniform algebra $C^*_u(G)$.
4. Exactness of the $C^*$-algebra $C^*_u(G)$.
5. KW-exactness of $G$.
6. Exactness of the reduced $C^*$-algebra $C^*_r(G)$.

This extends the classical equivalence known for discrete groups to a broad class of groupoids, provided they are inner amenable.

C. The Weak Containment Problem (WCP)

The paper resolves the WCP for specific classes of groupoids by linking it to exactness:

• Theorem 11.8: For a locally compact groupoid with a $T_0$ orbit space, amenability is equivalent to the Weak Containment Property (WCP) combined with inner exactness.
• Theorem 11.14: For étale groupoids, amenability is equivalent to the WCP combined with strong amenability at infinity.
• Corollary 11.19: If an étale groupoid admits a locally proper homomorphism into a KW-exact group, then WCP implies amenability.

These results explain why the HLS-groupoids (counterexamples to WCP $\implies$ amenability) fail: they lack inner exactness (or strong amenability at infinity).

4. Key Results and Examples

• HLS-Groupoids: The paper analyzes the Higson-Lafforgue-Skandalis (HLS) groupoids constructed from approximated groups. It proves that for these groupoids, inner exactness is equivalent to the amenability of the underlying group. Consequently, if the underlying group is non-amenable but exact, the HLS-groupoid is KW-exact but not inner exact, and thus fails the WCP $\implies$ amenability implication.
• Inverse Semigroups: The author connects the exactness of the reduced $C^*$-algebra of an inverse semigroup $S$ to the exactness of its maximal group homomorphic image $\Gamma_S$. If $S$ is strongly $E^*$-unitary, $C^*_r(S)$ is exact if and only if $\Gamma_S$ is exact.
• Transitive Groupoids: For transitive groupoids, KW-exactness is equivalent to amenability at infinity, and if the isotropy group is discrete, this is equivalent to $C^*$-exactness.

5. Significance and Open Questions

Significance:

• Unification: The work unifies the theory of exactness for discrete groups and locally compact groups into the broader framework of groupoids.
• Resolution of WCP: It provides a precise characterization of when the Weak Containment Property implies amenability, identifying inner exactness and strong amenability at infinity as the missing links in the general case.
• Counterexample Analysis: It clarifies the structural reasons (lack of inner exactness) why certain groupoids (like HLS-groupoids) violate the group-theoretic intuition regarding WCP.

Open Questions:

• Inner Amenability of Étale Groupoids: It is unknown whether all étale groupoids are inner amenable. If true, the equivalence of the six exactness conditions would hold for all étale groupoids.
• $C^*$-Exactness vs. KW-Exactness: For general locally compact groups, it remains open whether $C^*$-exactness implies KW-exactness.
• General WCP: Whether amenability is equivalent to (WCP + inner exactness) for all locally compact groupoids (not just those with $T_0$ orbit spaces) is an open problem.

Conclusion

This monograph is a definitive reference on the interplay between exactness, amenability, and the weak containment property in the context of locally compact groupoids. By introducing the concept of inner amenability for groupoids and utilizing fibrewise compactifications, the author successfully extends deep results from group theory to the groupoid setting, while clearly delineating the boundaries where these analogies break down.