Rigidity Conjectures

Abstract

We prove several rigidity results for corona $C^{\ast }$-algebras and Čech-Stone remainders under the assumption of Forcing Axioms. In particular, we prove that a strong version of Todorčević's $\OCA$ and Martin's Axiom at level ${\aleph }_1$ 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.

47 pages, to appear in Annales Scientifiques de l'École Normale Supérieure (ASENS)

Cited by in corpus (1)

• Forcing axioms and coronas of $C^{\ast }$-algebras