Isomorphism rigidity of uniform Roe algebras over arbitrary uniformly locally finite coarse spaces

Abstract

Let $(X,\mathcal{E})$ and $(Y,\mathcal{F})$ be uniformly locally finite coarse spaces. We prove that every $C^{\ast }$-algebra isomorphism $C_u^{\ast }(X,\mathcal{E})\cong C_u^{\ast }(Y,\mathcal{F})$ 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.

20 pages. All comments are welcome!