Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 19 (2024), 197 -- 215

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This work is licensed under a Creative Commons Attribution 4.0 International License.

ON SIMPLICITY OF CUNTZ ALGEBRAS AND ITS APPLICATIONS

Massoud Amini and Mahdi Moosazadeh

Abstract. Cuntz algebra 𝒪2 is the universal C*-algebra generated by two isometries s1, s2 satisfying s1s1*+s2s2*=1. This is separable, simple, infinite C*-algebra containing a copy of any nuclear C*-algebra. The C*-algebra 𝒪2 plays a central role in the modern theory of C*-algebras and appears in many fundamental statements, including a formulation of the celebrated Uniform Coefficient Theorem (UCT). There are several extensions of this notion, including Cuntz algebra 𝒪n, Cuntz-Krieger algebra ℱA for a matrix A, Cuntz-Pimsner algebra 𝒪X and its relaxation by Katsura for a C*-correspondence X, and Cuntz-Nica-Pimsner algebra 𝒩𝒪X, for a product system X. We give an overview of the construction of these classes of C*-algebras with a focus on conditions ensuring their simplicity, which is needed in the Elliott Classification Program, as it stands now. The results we present are now part of the literature, but we hope to shed a light on recent developments in a fascinating area of modern operator algebras.

2020 Mathematics Subject Classification: Primary 46L05; Secondary 46L08
Keywords: Cuntz algebra; Cuntz-Krieger algebra; Cuntz-Pimsner algebra; Cuntz-Nica-Pimsner algebra; Kishimoto condition; twisted crossed products; simplicity

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Massoud Amini
Tarbiat Modares University,
Department of Mathematics, Faculty of Mathematical Sciences, Tehran 14115-134, Iran.
e-mail: mamini@modares.ac.ir

Mahdi Moosazadeh
Tarbiat Modares University,
Department of Mathematics, Faculty of Mathematical Sciences, Tehran 14115-134, Iran.
e-mail: mahdimoosazadeh7@gmail.com

http://www.utgjiu.ro/math/sma