The notion of computable reducibility between equivalence relations on the natural numbers provides a natural computable analogue of Borel reducibility. We investigate the computable reducibility hierarchy, comparing and contrasting it with the Borel reducibility hierarchy from descriptive set theory. Meanwhile, the notion of computable reducibility appears well suited for an analysis of equivalence relations on the c.e. sets, and more specifically, on various classes of c.e. structures. This is a rich context with many natural examples, such as the isomorphism relation on c.e. graphs or on computably presented groups. Here, our exposition extends earlier work in the literature concerning the classification of computable structures. An abundance of open questions remains.
In 2007 Phillips and Weaver showed that, assuming the Continuum Hypothesis,
there exists an outer automorphism of the Calkin algebra. (The Calkin algebra
is the algebra of bounded operators on a separable complex Hilbert space,
modulo the compact operators.) In this paper we establish that the analogous
conclusion holds for a broad family of quotient algebras. Specifically, we will
show that assuming the Continuum Hypothesis, if $A$ is a separable algebra
which is either simple or stable, then the corona of $A$ has nontrivial
automorphisms. We also discuss a connection with cohomology theory, namely,
that our proof can be viewed as a computation of the cardinality of a
particular derived inverse limit
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