Lifting isomorphisms in $K$-theory through gradings of $C^*$-algebras
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913926729433088 |
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| author | Ruiz, Efren Sims, Aidan |
| author_facet | Ruiz, Efren Sims, Aidan |
| contents | We show that every strongly $\mathbb{Z}$-graded C*-algebra (equivalently, every C*-algebra carrying a strongly continuous $\mathbb{T}$-action with full spectral subspaces) is a Cuntz--Pimsner algebra, and describe subalgebras and subspaces that can be used as the coefficient algebra and module in the construction. We deduce that for surjective graded homomorphisms $ϕ$ of C*-algebras $A$ graded by torsion-free abelian groups $H$, if the restriction $ϕ_0$ of $ϕ$ to the zero-graded component $A_0$ of $A$ induces isomorphisms in K-theory, so does $ϕ$ itself. When $H$ is free abelian, we show how to pick out smaller subalgebras of $A_0$ on which it suffices to check that $ϕ$ induces isomorphisms in $K$-theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_03389 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lifting isomorphisms in $K$-theory through gradings of $C^*$-algebras Ruiz, Efren Sims, Aidan Operator Algebras 46L05 (primary), 46L80, 46L08 (secondary) We show that every strongly $\mathbb{Z}$-graded C*-algebra (equivalently, every C*-algebra carrying a strongly continuous $\mathbb{T}$-action with full spectral subspaces) is a Cuntz--Pimsner algebra, and describe subalgebras and subspaces that can be used as the coefficient algebra and module in the construction. We deduce that for surjective graded homomorphisms $ϕ$ of C*-algebras $A$ graded by torsion-free abelian groups $H$, if the restriction $ϕ_0$ of $ϕ$ to the zero-graded component $A_0$ of $A$ induces isomorphisms in K-theory, so does $ϕ$ itself. When $H$ is free abelian, we show how to pick out smaller subalgebras of $A_0$ on which it suffices to check that $ϕ$ induces isomorphisms in $K$-theory. |
| title | Lifting isomorphisms in $K$-theory through gradings of $C^*$-algebras |
| topic | Operator Algebras 46L05 (primary), 46L80, 46L08 (secondary) |
| url | https://arxiv.org/abs/2507.03389 |