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Shift equivalences through the lens of Cuntz–Krieger algebras

Toke Meier Carlsen, Adam Dor-On and Søren Eilers

Vol. 17 (2024), No. 1, 345–377

Motivated by Williams’ problem of measuring novel differences between shift equivalence (SE) and strong shift equivalence (SSE), we introduce three equivalence relations that provide new ways to obstruct SSE while merely assuming SE.

Our shift equivalence relations arise from studying graph C*-algebras, where a variety of intermediary equivalence relations naturally arise. As a consequence we realize a goal sought after by Muhly, Pask and Tomforde, measure a delicate difference between SSE and SE in terms of Pimsner dilations for C*-correspondences of adjacency matrices, and use this distinction to refute a proof from a previous paper.

shift equivalence, Williams' problem, Cuntz–Krieger algebras, Cuntz–Pimsner algebras, compatible shift equivalence, Pimsner dilations
Mathematical Subject Classification
Primary: 37A55, 46L55
Secondary: 37B10, 37A35, 46L08, 46L35
Received: 10 March 2022
Revised: 8 August 2022
Accepted: 15 September 2022
Published: 5 February 2024
Toke Meier Carlsen
Department of Mathematics
University of the Faroe Islands
Faroe Islands
Adam Dor-On
Mathematisches Institut
WWU Münster
Søren Eilers
Department of Mathematical Sciences
University of Copenhagen

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