Irreducibility and monicity for representations of k-graph C* -algebras

Carla Farsi; Elizabeth Gillaspy; Daniel Gonçalves

Irreducibility and monicity for representations of k-graph C* -algebras

Carla Farsi
, Elizabeth Gillaspy
, Daniel Gonçalves

Mathematical Sciences

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

2 Scopus citations

Abstract

The representations of a k-graph C∗ -algebra C∗ (Λ) which arise from Λ-semibranching function systems are closely linked to the dynamics of the k-graph Λ. In this paper, we undertake a systematic analysis of the question of irreducibility for these representations. We provide a variety of necessary and sufficient conditions for irreducibility, as well as a number of examples indicating the optimality of our results. In addition, we study the relationship between monic representations and the periodicity of Λ; our analysis yields results which are new even in the case of directed graphs. Finally, we explore the relationship between irreducible Λ-semibranching representations and purely atomic representations of C∗ (Λ). Throughout the paper, we work in the setting of row-finite source-free k-graphs; this paper constitutes the first analysis of Λ-semibranching representations at this level of generality.

Original language	English
Pages (from-to)	507-553
Number of pages	47
Journal	New York Journal of Mathematics
Volume	29
State	Published - 2023

Funding

Acknowledgments. C.F. was partially supported by the Simons Foundation Collaboration Grant for Mathematics #523991. E.G. was partially supported by the National Science Foundation (DMS-1800749). D.G. was partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), and by Capes-PrInt grant number 88881.310538/2018-01 - Brazil. C.F. also likes to thank the sabbatical program at the University of Colorado-Boulder for support. The authors thank Judith Packer for helpful conversations.

Funders	Funder number
DMS-1800749
Simons Foundation	523991
Conselho Nacional de Desenvolvimento Científico e Tecnológico	88881.310538/2018-01

Keywords

C∗-algebras
Monic representations
Row-finite higher-rank graphs
Λ-semibranching function systems

Cite this

@article{1cc6d4701b5b4e9080b8a82236a9912a,

title = "Irreducibility and monicity for representations of k-graph C* -algebras",

abstract = "The representations of a k-graph C∗ -algebra C∗ (Λ) which arise from Λ-semibranching function systems are closely linked to the dynamics of the k-graph Λ. In this paper, we undertake a systematic analysis of the question of irreducibility for these representations. We provide a variety of necessary and sufficient conditions for irreducibility, as well as a number of examples indicating the optimality of our results. In addition, we study the relationship between monic representations and the periodicity of Λ; our analysis yields results which are new even in the case of directed graphs. Finally, we explore the relationship between irreducible Λ-semibranching representations and purely atomic representations of C∗ (Λ). Throughout the paper, we work in the setting of row-finite source-free k-graphs; this paper constitutes the first analysis of Λ-semibranching representations at this level of generality.",

keywords = "C∗-algebras, Monic representations, Row-finite higher-rank graphs, Λ-semibranching function systems",

author = "Carla Farsi and Elizabeth Gillaspy and Daniel Gon{\c c}alves",

note = "Funding Information: Acknowledgments. C.F. was partially supported by the Simons Foundation Collaboration Grant for Mathematics \#523991. E.G. was partially supported by the National Science Foundation (DMS-1800749). D.G. was partially supported by Conselho Nacional de Desenvolvimento Cient{\'i}fico e Tecnol{\'o}gico (CNPq), and by Capes-PrInt grant number 88881.310538/2018-01 - Brazil. C.F. also likes to thank the sabbatical program at the University of Colorado-Boulder for support. The authors thank Judith Packer for helpful conversations. Publisher Copyright: {\textcopyright} 2023, University at Albany. All rights reserved.",

year = "2023",

language = "English",

volume = "29",

pages = "507--553",

journal = "New York Journal of Mathematics",

issn = "1076-9803",

publisher = "University at Albany",

}

TY - JOUR

T1 - Irreducibility and monicity for representations of k-graph C* -algebras

AU - Farsi, Carla

AU - Gillaspy, Elizabeth

AU - Gonçalves, Daniel

N1 - Funding Information: Acknowledgments. C.F. was partially supported by the Simons Foundation Collaboration Grant for Mathematics #523991. E.G. was partially supported by the National Science Foundation (DMS-1800749). D.G. was partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), and by Capes-PrInt grant number 88881.310538/2018-01 - Brazil. C.F. also likes to thank the sabbatical program at the University of Colorado-Boulder for support. The authors thank Judith Packer for helpful conversations. Publisher Copyright: © 2023, University at Albany. All rights reserved.

PY - 2023

Y1 - 2023

N2 - The representations of a k-graph C∗ -algebra C∗ (Λ) which arise from Λ-semibranching function systems are closely linked to the dynamics of the k-graph Λ. In this paper, we undertake a systematic analysis of the question of irreducibility for these representations. We provide a variety of necessary and sufficient conditions for irreducibility, as well as a number of examples indicating the optimality of our results. In addition, we study the relationship between monic representations and the periodicity of Λ; our analysis yields results which are new even in the case of directed graphs. Finally, we explore the relationship between irreducible Λ-semibranching representations and purely atomic representations of C∗ (Λ). Throughout the paper, we work in the setting of row-finite source-free k-graphs; this paper constitutes the first analysis of Λ-semibranching representations at this level of generality.

AB - The representations of a k-graph C∗ -algebra C∗ (Λ) which arise from Λ-semibranching function systems are closely linked to the dynamics of the k-graph Λ. In this paper, we undertake a systematic analysis of the question of irreducibility for these representations. We provide a variety of necessary and sufficient conditions for irreducibility, as well as a number of examples indicating the optimality of our results. In addition, we study the relationship between monic representations and the periodicity of Λ; our analysis yields results which are new even in the case of directed graphs. Finally, we explore the relationship between irreducible Λ-semibranching representations and purely atomic representations of C∗ (Λ). Throughout the paper, we work in the setting of row-finite source-free k-graphs; this paper constitutes the first analysis of Λ-semibranching representations at this level of generality.

KW - C∗-algebras

KW - Monic representations

KW - Row-finite higher-rank graphs

KW - Λ-semibranching function systems

UR - https://www.scopus.com/pages/publications/85164904016

M3 - Article

AN - SCOPUS:85164904016

SN - 1076-9803

VL - 29

SP - 507

EP - 553

JO - New York Journal of Mathematics

JF - New York Journal of Mathematics

ER -

Irreducibility and monicity for representations of k-graph C* -algebras

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