On the Nielsen-Thomsen sequence
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866918534714490880 |
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| author | Cantier, Laurent |
| author_facet | Cantier, Laurent |
| contents | The Nielsen-Thomsen sequence plays a pivotal role in refining invariants for C$^*$-algebras beyond the Elliott classification framework. This paper revisits the sequence, introducing the concepts of Nielsen-Thomsen bases, rotation maps and diagonalisable morphisms, to better understand its unnatural splitting. These insights enable novel comparison methods for *-homomorphisms at the level of the Hausdorffized algebraic K$_1$-groups, and subsequently the Hausdorffized unitary Cuntz group. We apply our methods to classification via the Hausdorffized unitary Cuntz semigroup. In particular, we present a new proof of the non-isomorphism between two A$\mathbb{T}$-algebras constructed by Gong, Jiang and Li. We also exhibit several pairs of non-unitarily equivalent *-homomorphisms with domain C($\mathbb{T}$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_11975 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Nielsen-Thomsen sequence Cantier, Laurent Operator Algebras The Nielsen-Thomsen sequence plays a pivotal role in refining invariants for C$^*$-algebras beyond the Elliott classification framework. This paper revisits the sequence, introducing the concepts of Nielsen-Thomsen bases, rotation maps and diagonalisable morphisms, to better understand its unnatural splitting. These insights enable novel comparison methods for *-homomorphisms at the level of the Hausdorffized algebraic K$_1$-groups, and subsequently the Hausdorffized unitary Cuntz group. We apply our methods to classification via the Hausdorffized unitary Cuntz semigroup. In particular, we present a new proof of the non-isomorphism between two A$\mathbb{T}$-algebras constructed by Gong, Jiang and Li. We also exhibit several pairs of non-unitarily equivalent *-homomorphisms with domain C($\mathbb{T}$). |
| title | On the Nielsen-Thomsen sequence |
| topic | Operator Algebras |
| url | https://arxiv.org/abs/2412.11975 |