Amenable unitary representations of locally compact groupoids

Let $G$ be a second countable locally compact groupoid equipped with a Haar system $λ$.In this work, we introduce and develop the notion of amenability for continuous unitary representations of $G$, formulated in terms of Hilbert bundles over the unit space $G^{0}$. We prove that $G$ is amenable if and only if its left regular representation is amenable, thereby extending Bekka’s characterisation of amenable unitary representations from groups to groupoids. We further investigate the amenability of induced representations of $G$ and also study the representation of properly amenable groupoids. Finally, we define a topological invariant mean associated with a representation, constructed by utilising the theory of operator-valued vector measures on the unit space $G^{0}$, to characterise amenability.

💡 Research Summary

The paper develops a theory of amenability for continuous unitary representations of second‑countable locally compact groupoids equipped with a Haar system. Using the framework of Hilbert bundles over the unit space, the authors define a representation (\pi) to be amenable if there exists a net of Hilbert‑Schmidt operators ({T_i}) on the Hilbert (C_0(G^0))-module of continuous sections such that (i) the fibrewise Hilbert‑Schmidt norms approach 1 uniformly on compact subsets of the unit space, and (ii) the inner products (\langle x\cdot T_i d(x), T_i r(x)\rangle) converge to 1 uniformly on compact subsets of the groupoid. This definition mirrors Renault’s notion of amenability for the groupoid itself and extends Bekka’s characterization for groups.

The authors first recall necessary background on groupoids, Haar systems, and Schatten‑class operators on countably generated Hilbert (C^*)-modules. They then construct the left regular representation of the groupoid on the bundle of Hilbert‑Schmidt operators and show that the space of finite‑rank adjointable operators is dense in the Hilbert‑Schmidt module, allowing the use of approximation arguments similar to those in the group case.

Key results include:

1. Stability properties: Unitarily equivalent representations and the conjugate representation of an amenable representation are themselves amenable.

2. Equivalence of amenability notions (Theorem 3.6): For a second‑countable locally compact groupoid (G), the following are equivalent:

   • (i) (G) is amenable (in the sense of Renault).
   • (ii) Every continuous unitary representation of (G) is amenable.
   • (iii) The left regular representation (\lambda_G) is amenable.

   The proof of (i)⇒(ii) uses an approximate invariant mean on the groupoid to build a net of positive trace‑class operators that satisfy the definition. The implication (iii)⇒(i) extracts a topological invariant mean from the amenability of the regular representation, thereby recovering the classical Bekka theorem for groups as a special case.

3. Induced representations: If (H) is a closed wide subgroupoid of (G) and (\sigma) a representation of (H), then the induced representation (\operatorname{ind}_G^H(\sigma)) inherits amenability from (\sigma). Moreover, restriction of an amenable representation to any closed subgroupoid remains amenable, and restriction to isotropy groups is also amenable.

4. Proper amenability: The paper introduces a stronger notion, “properly amenable” groupoids, characterized by the existence of a non‑zero equivariant family of trace‑class operators ({K_u}_{u\in G^0}) whose trace function lies in (C_b(G^0)). This condition is shown to be equivalent to the existence of a certain invariant kernel for every representation.

5. Topological invariant mean: Leveraging the theory of operator‑valued vector measures on (G^0), the authors define a topological invariant mean associated with a representation. This mean provides an alternative, measure‑theoretic characterization of amenability and connects the representation‑theoretic perspective with the classical invariant mean on the groupoid.

Throughout, the analysis relies heavily on the machinery of Hilbert (C^*)-modules, particularly the Schatten classes (L_1) and (L_2), and on the continuity properties of the bundle of trace‑class operators. The paper also discusses the dual space structure of these bundles, enabling the construction of the invariant mean.

In conclusion, the work successfully generalizes Bekka’s amenability criteria from groups to locally compact groupoids, establishes the equivalence between groupoid amenability and amenability of its regular representation, and provides new tools—induced representations, proper amenability, and topological invariant means—that broaden the scope of amenability studies in non‑commutative harmonic analysis and groupoid (C^*)-algebra theory.