K-Theory and Structural Properties of $C^*$-Algebras Associated with Relative Generalized Boolean Dynamical Systems
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| Format: | Preprint |
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2025
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| _version_ | 1866915496631205888 |
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| author | Carlsen, Toke Meier Kang, Eun Ji |
| author_facet | Carlsen, Toke Meier Kang, Eun Ji |
| contents | 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 $K_1$-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 $K_0$-liftable. Furthermore, we deduce that if $C^*(\CB, \CL, θ, \CI_\af; \CJ)$ is separable and purely infinite, then it has real rank zero. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_12738 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | K-Theory and Structural Properties of $C^*$-Algebras Associated with Relative Generalized Boolean Dynamical Systems Carlsen, Toke Meier Kang, Eun Ji Operator Algebras 46L05, 46L55, 46L80, 46L85 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 $K_1$-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 $K_0$-liftable. Furthermore, we deduce that if $C^*(\CB, \CL, θ, \CI_\af; \CJ)$ is separable and purely infinite, then it has real rank zero. |
| title | K-Theory and Structural Properties of $C^*$-Algebras Associated with Relative Generalized Boolean Dynamical Systems |
| topic | Operator Algebras 46L05, 46L55, 46L80, 46L85 |
| url | https://arxiv.org/abs/2509.12738 |