5 problems

Refined embedding conjecture. The only von Neumann algebras that can embed into

Folk conjecture. The only von Neumann algebras that can embed into a uniform Roe algebra or a quasi-local algebra associated to a uniformly locally finite metric space are those of…

Uniform Roe algebra rigidity conjecture. The $C^*$-algebras $C^*_u(X)$ and $C^*_u(Y)$ are Morita equivalent (respectively, isomorphic) if and only if $(X,d)$ and $(Y,\partial)$ are…

Rough Baum–Connes conjecture. The map $\mu_u$ is an isomorphism. This conjecture asserts that uniform $K$-homology of the Rips complexes computes the $K$-theory of the uniform Roe…

Let $X$ be a bounded geometry metric space, and let $C^*_u(X)$ denote its uniform Roe algebra. The real-rank conjecture. If $X$ has asymptotic dimension at least two, then…