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arXiv · 2412.11975

On the Nielsen-Thomsen sequence

Laurent Cantier
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Abstract

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($\mathbb{T}$).

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Laurent Cantier. 2024-12-16. On the Nielsen-Thomsen sequence. https://arxiv.org/abs/2412.11975

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