Rigidity Conjectures

We prove several rigidity results for corona $C^{\ast }$-algebras and \v{C}ech-Stone remainders under the assumption of Forcing Axioms. In particular, we prove that a strong version of Todor\v{c}evi'c’s $\text{OCA}$ and Martin’s Axiom at level ${\mathrm{\aleph }}_1$ imply: (i) that if $X$ and $Y$ are locally compact second countable topological spaces, then all homeomorphisms between $\beta X\setminus X$ and $\beta Y\setminus Y$ are induced by homeomorphisms between cocompact subspaces of $X$ and $Y$; (ii) that all automorphisms of the corona algebra of a separable $C^{\ast }$-algebra are trivial in a topological sense; (iii) that if $A$ is a unital separable infinite-dimensional $C^{\ast }$-algebra, the corona algebra of $A\otimes \mathcal{K}(H)$ does not embed into the Calkin algebra. All these results do not hold under the Continuum Hypothesis.