On Conjugacy of Subalgebras of Graph C*-algebras

Sophie Emma Mikkelsen; Wojciech Szymanski; Jeong Hee Hong; Tomohiro Hayashi

doi:10.1007/978-3-030-34072-8_17

On Conjugacy of Subalgebras of Graph C*-algebras

Sophie Emma Mikkelsen, Wojciech Szymanski, Jeong Hee Hong, Tomohiro Hayashi

Department of Mathematics and Computer Science

Nagoya Institute of Technology

Research output: Chapter in Book/Report/Conference proceeding › Article in proceedings › Research › peer-review

Abstract

The problem of inner vs outer conjugacy of subalgebras of certain graph C*-algebras is investigated. For a large class of finite graphs E, we show that whenever α is a vertex-fixing quasi-free automorphism of the corresponding graph C*-algebra C* (E) such that α(DE) ≠ DE where DE is the canonical MASA in C*(E), then α(DE) ≠ ωDEω* for all unitaries ω ∈ C* (E). That is, the two MASAs DE and α(DE) of C*(E) are outer but not inner conjugate. For the Cuntz algebras On, we find a criterion which guarantees that a polynomial automorphism moves the canonical UHF subalgebra to a non-inner conjugate UHF subalgebra. The criterion is phrased in terms of rescaling of trace on diagonal projections.

Original language	English
Title of host publication	Geometric Methods in Physics XXXVII : Workshop 2018
Publisher	Birkhäuser Verlag
Publication date	2019
Pages	163-173
ISBN (Print)	978-3-030-34071-1
ISBN (Electronic)	978-3-030-34072-8
DOIs	https://doi.org/10.1007/978-3-030-34072-8_17
Publication status	Published - 2019
Event	XXXIX Workshop on Geometric Methods in Physics - Duration: 1. Jul 2018 → 7. Jul 2018

Conference

Conference	XXXIX Workshop on Geometric Methods in Physics
Period	01/07/2018 → 07/07/2018

Series	Trends in Mathematics

Keywords

C-functions
Cuntz algebra
MASA
S-functions
UHF-subalgebra
automorphism
graph C -algebra
inner and outer conjugacy

Documents & Links

10.1007/978-3-030-34072-8_17

1 Ph.D. thesis

Structure and symmetries in C*-algebras
Mikkelsen, S. E., 30. Nov 2021, Syddansk Universitet. Det Naturvidenskabelige Fakultet. 227 p.
Research output: Thesis › Ph.D. thesis

Open Access
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Cite this

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title = "On Conjugacy of Subalgebras of Graph C*-algebras",

abstract = "The problem of inner vs outer conjugacy of subalgebras of certain graph C*-algebras is investigated. For a large class of finite graphs E, we show that whenever α is a vertex-fixing quasi-free automorphism of the corresponding graph C*-algebra C* (E) such that α(DE) ≠ DE where DE is the canonical MASA in C*(E), then α(DE) ≠ ωDEω* for all unitaries ω ∈ C* (E). That is, the two MASAs DE and α(DE) of C*(E) are outer but not inner conjugate. For the Cuntz algebras On, we find a criterion which guarantees that a polynomial automorphism moves the canonical UHF subalgebra to a non-inner conjugate UHF subalgebra. The criterion is phrased in terms of rescaling of trace on diagonal projections.",

keywords = "C-functions, Cuntz algebra, MASA, S-functions, UHF-subalgebra, automorphism, graph C -algebra, inner and outer conjugacy",

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T1 - On Conjugacy of Subalgebras of Graph C*-algebras

AU - Mikkelsen, Sophie Emma

AU - Szymanski, Wojciech

AU - Hong, Jeong Hee

AU - Hayashi, Tomohiro

PY - 2019

Y1 - 2019

N2 - The problem of inner vs outer conjugacy of subalgebras of certain graph C*-algebras is investigated. For a large class of finite graphs E, we show that whenever α is a vertex-fixing quasi-free automorphism of the corresponding graph C*-algebra C* (E) such that α(DE) ≠ DE where DE is the canonical MASA in C*(E), then α(DE) ≠ ωDEω* for all unitaries ω ∈ C* (E). That is, the two MASAs DE and α(DE) of C*(E) are outer but not inner conjugate. For the Cuntz algebras On, we find a criterion which guarantees that a polynomial automorphism moves the canonical UHF subalgebra to a non-inner conjugate UHF subalgebra. The criterion is phrased in terms of rescaling of trace on diagonal projections.

AB - The problem of inner vs outer conjugacy of subalgebras of certain graph C*-algebras is investigated. For a large class of finite graphs E, we show that whenever α is a vertex-fixing quasi-free automorphism of the corresponding graph C*-algebra C* (E) such that α(DE) ≠ DE where DE is the canonical MASA in C*(E), then α(DE) ≠ ωDEω* for all unitaries ω ∈ C* (E). That is, the two MASAs DE and α(DE) of C*(E) are outer but not inner conjugate. For the Cuntz algebras On, we find a criterion which guarantees that a polynomial automorphism moves the canonical UHF subalgebra to a non-inner conjugate UHF subalgebra. The criterion is phrased in terms of rescaling of trace on diagonal projections.

KW - C-functions

KW - Cuntz algebra

KW - MASA

KW - S-functions

KW - UHF-subalgebra

KW - automorphism

KW - graph C -algebra

KW - inner and outer conjugacy

U2 - 10.1007/978-3-030-34072-8_17

DO - 10.1007/978-3-030-34072-8_17

M3 - Article in proceedings

SN - 978-3-030-34071-1

T3 - Trends in Mathematics

SP - 163

EP - 173

BT - Geometric Methods in Physics XXXVII :

PB - Birkhäuser Verlag

T2 - XXXIX Workshop on Geometric Methods in Physics

Y2 - 1 July 2018 through 7 July 2018

ER -

On Conjugacy of Subalgebras of Graph C*-algebras

Abstract

Conference

Keywords

Documents & Links

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Related Research Output

Structure and symmetries in C*-algebras

Cite this