Year: 2018
Type: preprint
Abstract: We prove several rigidity results for corona $C^*$-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 $βX\setminus X$ and $βY\setminus Y$ are induced by homeomorphisms between cocompact subspaces of $X$ and $Y$; (ii) that all automorphisms of the corona algebra of a separable $C^*$-algebra are trivial in a topological sense; (iii) that if $A$ is a unital separable infinite-dimensional $C^*$-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.
Source: arXiv (Cornell University)
Author: Vignati, Alessandro
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2903551078