Rigidity of ${\ell }^p$ Roe-type algebras

Abstract

We investigate the rigidity of the ${\ell }^p$ analog of Roe-type algebras. In particular, we show that if $p\in [1,\infty )\setminus \{2\}$, then an isometric isomorphism between the ${\ell }^p$ uniform Roe algebras of two metric spaces with bounded geometry yields a bijective coarse equivalence between the underlying metric spaces, while a stable isometric isomorphism yields a coarse equivalence. We also obtain similar results for other ${\ell }^p$ Roe-type algebras. In this paper, we do not assume that the metric spaces have Yu's property A or finite decomposition complexity.

arXiv commentFinal version, 19 pages. To appear in Bulletin of the London Mathematical Society. Corrected minor typos, and updated bibliography

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Cited by in corpus (9)

• Uniform Roe algebras of uniformly locally finite metric spaces are rigid
• Rigidity results for $L^p$-operator algebras and applications
• A quasi-local characterisation of $L^p$-Roe algebras
• On the uniform Roe algebra as a Banach algebra and embeddings of ${\ell }_p$ uniform Roe algebras
• Dynamical Complexity and $K$-Theory of $L^p$ Operator Crossed Products
• Banach algebras associated to twisted étale groupoids: inverse semigroup disintegration and representations on $L^p$-spaces
• Persistence approximation property for $L^p$ operator algebras
• $L^p$-spectral triples and $p$-quantum compact metric spaces
• $L^p$ coarse Baum-Connes conjecture via $C_0$ coarse geometry