The rigidity problem for uniform Roe algebras

We solve the rigidity problem for uniform Roe algebras, by showing that two uniformly locally finite metric spaces with isomorphic uniform Roe algebras are bijectively coarsely equivalent.

💡 Research Summary

The paper addresses the long‑standing rigidity problem for uniform Roe algebras. Given two uniformly locally finite (u.l.f.) metric spaces (X) and (Y), the author proves that any ()-isomorphism between their uniform Roe algebras (C^{}{u}(X)) and (C^{*}{u}(Y)) forces the underlying spaces to be bijectively coarsely equivalent. This result removes all previously required geometric hypotheses such as Property A or the Operator Norm Localization (ONL) property, thereby establishing the strongest possible correspondence between the algebraic structure of uniform Roe algebras and the large‑scale geometry of the spaces they encode.

The proof proceeds by exploiting the relationship between uniform Roe algebras and the Higson corona. A ()-isomorphism (\Phi : C^{}{u}(X) \to C^{*}{u}(Y)) induces an isomorphism of the Higson coronas (C_{\nu}(X)) and (C_{\nu}(Y)). Since the Higson corona is the centre of the uniform Roe corona, (\Phi) maps slowly oscillating functions on (X) to compact perturbations of slowly oscillating functions on (Y). By a careful diagonalisation argument (inspired by recent analyses of Higson coronas), the author shows that “flattened” indicator functions in (\ell^{\infty}(X)) are sent arbitrarily close to (\ell^{\infty}(Y)). This key observation yields a norm estimate that replaces the earlier reliance on Property A or ONL.

Using this estimate, the author constructs a set‑valued map (\alpha : X \to \mathrm{Fin}(Y)) with the property that any choice function (f) satisfying (f(x) \in \alpha(x)) for all (x) is a coarse embedding. Hall’s marriage theorem then provides an injective coarse map (f : X \to Y). Applying the same construction to (\Phi^{-1}) yields an injective coarse map (g : Y \to X). The maps are shown to be mutual coarse inverses, and a Cantor–Schröder–Bernstein argument upgrades them to a bijective coarse equivalence.

Beyond the main rigidity theorem (Theorem A), the paper establishes two further structural results. Theorem B proves that any Roe Cartan subalgebra (A \subset C^{}_{u}(X)) is unitarily conjugate to the canonical Cartan masa (\ell^{\infty}(X)), without assuming Property A. This gives a strong uniqueness statement for Cartan subalgebras in uniform Roe algebras. Theorem C shows that the natural homomorphism (\mathrm{BijCoa}(X) \to \mathrm{Out}(C^{}_{u}(X))) is an isomorphism for every u.l.f. space (X). Consequently, every outer automorphism of the uniform Roe algebra arises from a bijective coarse equivalence of the underlying space, and vice versa.

The paper’s methodology—leveraging the Higson corona and the algebra of slowly oscillating functions—opens new avenues for studying other rigidity‑type questions. The author suggests that similar techniques could be applied to investigate coarse embeddings between spaces via embeddings of uniform Roe algebras, as well as to explore dimension theory for Roe algebras. The work thus not only resolves the rigidity problem in its full generality but also provides a versatile toolkit for future research at the interface of coarse geometry and operator algebras.