Extensions of C*-algebras
James Gabe, Huaxin Lin, Ping Wong Ng

TL;DR
This paper classifies essential extensions of certain C*-algebras using KK theory, characterizes liftability and equivalence relations, and extends the Voiculescu theorem to new classes of C*-algebras.
Contribution
It provides a KK-theoretic classification of essential extensions of separable amenable C*-algebras by simple C*-algebras with continuous scale, including liftability criteria and a noncommutative Weyl--von Neumann theorem.
Findings
Classification of extensions via KK theory.
Criteria for liftability of extensions.
A noncommutative Weyl--von Neumann theorem for certain C*-algebras.
Abstract
Let be a separable amenable -algebra and a non-unital and -unital simple -algebra with continuous scale ( need not be stable). We classify, up to unitary equivalence, all essential extensions of the form using KK theory. There are characterizations of when the relation of weak unitary equivalence is the same as the relation of unitary equivalence, and characterizations of when an extension is liftable (a.k.a.~trivial or split). In the case where is purely infinite, an essential extension is liftable if and only if in . When is stably finite, the extension is often not liftable when in Finally, when additionally has tracial rank zero and when belongs to a sufficiently regular class of unital…
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Taxonomy
TopicsAdvanced Operator Algebra Research · Lanthanide and Transition Metal Complexes · Holomorphic and Operator Theory
