Anomalous symmetries of classifiable C*-algebras
<p>This thesis studies the existence and uniqueness of $G$-kernels on those C$^*$-algebras classified by the Elliott programme. We develop two obstructions to the possible $H^3$ invariants of a $G$-kernel. These obstructions arise from studying the unitary algebraic $K_1$ group and the topolog...
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| Format: | Thesis |
| Language: | English |
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2023
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| _version_ | 1797110824024670208 |
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| author | Girón Pacheco, S |
| author2 | White, S |
| author_facet | White, S Girón Pacheco, S |
| author_sort | Girón Pacheco, S |
| collection | OXFORD |
| description | <p>This thesis studies the existence and uniqueness of $G$-kernels on those C$^*$-algebras classified by the Elliott programme. We develop two obstructions to the possible $H^3$ invariants of a $G$-kernel. These obstructions arise from studying the unitary algebraic $K_1$ group and the topological $K_0$ group of a C$^*$-algebra. As a consequence of these obstructions, we show that any $G$-kernel on the Jiang-Su algebra has trivial $H^3$ invariant. Similarly, for finite groups $G$, any $G$-kernel on the Cuntz algebra $\mathcal{O}_\infty$ must have trivial $H^3$ invariant.</p>
<p>We construct multiple examples of $G$-kernels with non-trivial $H^3$ invariant and, under a UHF-absorption condition, we classify those $G$-kernels that have the Rokhlin property on both Kirchberg algebras satisfying the UCT and unital, separable, simple, nuclear, tracially AF C$^*$-algebras that satisfy the UCT. As a follow up to this classification, we study the structure of $G$-kernels with the Rokhlin property.</p> |
| first_indexed | 2024-03-07T08:00:09Z |
| format | Thesis |
| id | oxford-uuid:34239bc3-1111-46ff-940f-ffe327d0a7aa |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T08:00:09Z |
| publishDate | 2023 |
| record_format | dspace |
| spelling | oxford-uuid:34239bc3-1111-46ff-940f-ffe327d0a7aa2023-09-28T12:56:57ZAnomalous symmetries of classifiable C*-algebrasThesishttp://purl.org/coar/resource_type/c_db06uuid:34239bc3-1111-46ff-940f-ffe327d0a7aaHigher categoriesOperator algebrasK-theoryEnglishHyrax Deposit2023Girón Pacheco, SWhite, SEvington, S<p>This thesis studies the existence and uniqueness of $G$-kernels on those C$^*$-algebras classified by the Elliott programme. We develop two obstructions to the possible $H^3$ invariants of a $G$-kernel. These obstructions arise from studying the unitary algebraic $K_1$ group and the topological $K_0$ group of a C$^*$-algebra. As a consequence of these obstructions, we show that any $G$-kernel on the Jiang-Su algebra has trivial $H^3$ invariant. Similarly, for finite groups $G$, any $G$-kernel on the Cuntz algebra $\mathcal{O}_\infty$ must have trivial $H^3$ invariant.</p> <p>We construct multiple examples of $G$-kernels with non-trivial $H^3$ invariant and, under a UHF-absorption condition, we classify those $G$-kernels that have the Rokhlin property on both Kirchberg algebras satisfying the UCT and unital, separable, simple, nuclear, tracially AF C$^*$-algebras that satisfy the UCT. As a follow up to this classification, we study the structure of $G$-kernels with the Rokhlin property.</p> |
| spellingShingle | Higher categories Operator algebras K-theory Girón Pacheco, S Anomalous symmetries of classifiable C*-algebras |
| title | Anomalous symmetries of classifiable C*-algebras |
| title_full | Anomalous symmetries of classifiable C*-algebras |
| title_fullStr | Anomalous symmetries of classifiable C*-algebras |
| title_full_unstemmed | Anomalous symmetries of classifiable C*-algebras |
| title_short | Anomalous symmetries of classifiable C*-algebras |
| title_sort | anomalous symmetries of classifiable c algebras |
| topic | Higher categories Operator algebras K-theory |
| work_keys_str_mv | AT gironpachecos anomaloussymmetriesofclassifiablecalgebras |