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Classification of irreversible and reversible Pimsner operator algebras

Part of: Linear spaces and algebras of operators Selfadjoint operator algebras

Published online by Cambridge University Press:  13 January 2021

Adam Dor-On ,
Søren Eilers  and
Shirly Geffen
Adam Dor-On
Affiliation:
Department of Mathematical Sciences, University of Copenhagen, Universitetspark 5, 2100Copenhagen, Denmarkadoron@math.ku.dk
Søren Eilers
Affiliation:
Department of Mathematical Sciences, University of Copenhagen, Universitetspark 5, 2100Copenhagen, Denmarkeilers@math.ku.dk
Shirly Geffen
Affiliation:
Department of Mathematics, Ben-Gurion University of the Negev, PO box 653, Be'er Sheva, 8410501Israelshirlyg@post.bgu.ac.il
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Abstract

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.

Keywords

classificationtensor algebrasPimsner algebrasrigiditynon-commutative boundaryK-theorygraph algebrasreconstruction

MSC classification

Primary: 47L30: Abstract operator algebras on Hilbert spaces 46L35: Classifications of $C^*$-algebras
Secondary: 47L55: Representations of (nonselfadjoint) operator algebras 46L80: $K$-theory and operator algebras (including cyclic theory) 46L08: $C^*$-modules
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 12 , December 2020 , pp. 2510 - 2535
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Copyright
© The Author(s) 2021

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Footnotes

The first author was supported by NSF grant DMS-1900916 and by the European Union's Horizon 2020 Marie Sklodowska-Curie grant No 839412. The second author was supported by the DFF-Research Project 2 ‘Automorphisms and Invariants of Operator Algebras’, no. 7014-00145B, and by the DNRF through the Centre for Symmetry and Deformation (DNRF92). The third author was supported by a Negev fellowship, a Minerva fellowship programme, an ISF grant no. 476/16 and the DFG through SFB 878 and EXC 2044 Mathematics Münster: Dynamics–Geometry–Structure.

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