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Classification of irreversible and reversible Pimsner operator algebras
- Part of
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- Journal:
- Compositio Mathematica / Volume 156 / Issue 12 / December 2020
- Published online by Cambridge University Press:
- 13 January 2021, pp. 2510-2535
- Print publication:
- December 2020
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- Article
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Since their inception in the 1930s by von Neumann, operator algebras have been used to shed light on many mathematical theories. Classification results for self-adjoint and non-self-adjoint operator algebras manifest this approach, but a clear connection between the two has been sought since their emergence in the late 1960s. We connect these seemingly separate types of results by uncovering a hierarchy of classification for non-self-adjoint operator algebras and
$C^{*}$-algebras with additional 
$C^{*}$-algebraic structure. Our approach naturally applies to algebras arising from 
$C^{*}$-correspondences to resolve self-adjoint and non-self-adjoint isomorphism problems in the literature. We apply our strategy to completely elucidate this newly found hierarchy for operator algebras arising from directed graphs.
