Homotopy of Product Systems and K-Theory of Cuntz-Nica-Pimsner Algebras

James Fletcher; Elizabeth Gillaspy; Aidan Sims

doi:10.1512/iumj.2022.71.8810

Homotopy of Product Systems and K-Theory of Cuntz-Nica-Pimsner Algebras

James Fletcher
, Elizabeth Gillaspy
, Aidan Sims

Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

Abstract

We introduce the notion of a homotopy of product systems, and show that the Cuntz-Nica-Pimsner algebras of homotopic product systems over N^k have isomorphic K-theory. As an application, we give a new proof that the K-theory of a 2-graph C^*-algebra is independent of the factorisation rules, and we further show that the K-theory of any twisted 2-graph C^*-algebra is independent of the twisting 2-cocycle. We also explore applications to K-theory for the C^*-algebras of single-vertex k-graphs, reducing the question of whether the K-theory is independent of the factorisation rules to a question about path-connectedness of the space of solutions to an equation of Yang-Baxter type.

Original language	English
Pages (from-to)	307-338
Number of pages	32
Journal	Indiana University Mathematics Journal
Volume	71
Issue number	5
DOIs	https://doi.org/10.1512/iumj.2022.71.8810
State	Published - 2022

Funding

4.3. Acknowledgements. The authors wish to thank the anonymous referee for helpful comments and suggestions. This research was supported in part by the Royal Society of New Zealand (Marsden grant no. 15-UOO-071), the Small Grants Program of the University of Montana, the US National Science Foundation (grant no. DMS-1800749 to E.G.), and the Australian Research Council (grant no. DP180100595).

Funders	Funder number
DMS-1800749
Australian Research Council	DP180100595
15-UOO-071

Keywords

Cuntz-Nica-Pimsner
K-theory
Product system
higher-rank graph

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10.1512/iumj.2022.71.8810

Cite this

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title = "Homotopy of Product Systems and K-Theory of Cuntz-Nica-Pimsner Algebras",

abstract = "We introduce the notion of a homotopy of product systems, and show that the Cuntz-Nica-Pimsner algebras of homotopic product systems over Nk have isomorphic K-theory. As an application, we give a new proof that the K-theory of a 2-graph C*-algebra is independent of the factorisation rules, and we further show that the K-theory of any twisted 2-graph C*-algebra is independent of the twisting 2-cocycle. We also explore applications to K-theory for the C*-algebras of single-vertex k-graphs, reducing the question of whether the K-theory is independent of the factorisation rules to a question about path-connectedness of the space of solutions to an equation of Yang-Baxter type.",

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author = "James Fletcher and Elizabeth Gillaspy and Aidan Sims",

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AU - Fletcher, James

AU - Gillaspy, Elizabeth

AU - Sims, Aidan

N1 - Publisher Copyright: Indiana University Mathematics Journal ©

PY - 2022

Y1 - 2022

N2 - We introduce the notion of a homotopy of product systems, and show that the Cuntz-Nica-Pimsner algebras of homotopic product systems over Nk have isomorphic K-theory. As an application, we give a new proof that the K-theory of a 2-graph C*-algebra is independent of the factorisation rules, and we further show that the K-theory of any twisted 2-graph C*-algebra is independent of the twisting 2-cocycle. We also explore applications to K-theory for the C*-algebras of single-vertex k-graphs, reducing the question of whether the K-theory is independent of the factorisation rules to a question about path-connectedness of the space of solutions to an equation of Yang-Baxter type.

AB - We introduce the notion of a homotopy of product systems, and show that the Cuntz-Nica-Pimsner algebras of homotopic product systems over Nk have isomorphic K-theory. As an application, we give a new proof that the K-theory of a 2-graph C*-algebra is independent of the factorisation rules, and we further show that the K-theory of any twisted 2-graph C*-algebra is independent of the twisting 2-cocycle. We also explore applications to K-theory for the C*-algebras of single-vertex k-graphs, reducing the question of whether the K-theory is independent of the factorisation rules to a question about path-connectedness of the space of solutions to an equation of Yang-Baxter type.

KW - Cuntz-Nica-Pimsner

KW - K-theory

KW - Product system

KW - higher-rank graph

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DO - 10.1512/iumj.2022.71.8810

M3 - Article

AN - SCOPUS:85148005358

SN - 0022-2518

VL - 71

SP - 307

EP - 338

JO - Indiana University Mathematics Journal

JF - Indiana University Mathematics Journal

IS - 5

ER -

Homotopy of Product Systems and K-Theory of Cuntz-Nica-Pimsner Algebras

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