C_u^*(X) can be a hereditary corner of C_u^*(Y) with no coarse embedding X→Y; sparse compact-ghost targets restore injective coarse embeddability.
Isomorphism rigidity of uniform Roe algebras over arbitrary uniformly locally finite coarse spaces
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abstract
Let $(X,\mathcal E)$ and $(Y,\mathcal F)$ be uniformly locally finite coarse spaces. We prove that every $C^*$-algebra isomorphism $C_u^*(X,\mathcal E)\cong C_u^*(Y,\mathcal F)$ forces $(X,\mathcal E)$ and $(Y,\mathcal F)$ to be bijectively coarsely equivalent. This completely resolves the isomorphism rigidity problem for uniform Roe algebras over arbitrary uniformly locally finite coarse spaces.
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On the embedding rigidity problem for uniformly locally finite coarse spaces
C_u^*(X) can be a hereditary corner of C_u^*(Y) with no coarse embedding X→Y; sparse compact-ghost targets restore injective coarse embeddability.