K-Theory and Structural Properties of C^*-Algebras Associated with Relative Generalized Boolean Dynamical Systems

2025/09/16 by Toke Meier Carlsen, Carlsen, Toke Meier, Eun Ji Kang +1
Computer Science · Mathematics · #46L05 #46L55 #46L80 #46L85 #Advanced Algebra and Logic #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #FOS: Mathematics #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.2509.12738

openalex publication_date 2025/09/16 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28

Abstract

We present an explicit formula for the K-theory of the C^*-algebra associated with a relative generalized Boolean dynamical system (\CB, \CL, θ, \CI_\af; \CJ). In particular, we find concrete generators for the K₁-group of C^*(\CB, \CL, θ, \CI_\af; \CJ). We also prove that every gauge-invariant ideal of C^*(\CB, \CL, θ, \CI_\af; \CJ) is Morita equivalent to a C^*-algebra of a relative generalized Boolean dynamical system. As a structural application, we show that if the underlying Boolean dynamical system (\CB, \CL, θ) satisfies Condition (K), then the associated C^*-algebra is K₀-liftable. Furthermore, we deduce that if C^*(\CB, \CL, θ, \CI_\af; \CJ) is separable and purely infinite, then it has real rank zero.

Citations

• A Cuntz--Krieger Uniqueness theorem for C*-algebras of relative generalized Boolean dynamical systems
• On C^*-algebras associated with C^*-correspondences

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• C*-algebras of generalized Boolean dynamical systems as partial crossed products
• A Cuntz--Krieger Uniqueness theorem for C*-algebras of relative generalized Boolean dynamical systems
• Relative Generalized Boolean Dynamical System Algebras
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• Ideal structure of C*-algebras of singly generated dynamical systems