On the Nielsen-Thomsen sequence

2024/12/16 by Cantier, Laurent
#FOS: Mathematics #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.2412.11975

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 K₁-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 the two A\mathbbT-algebras of [19]. We also exhibit several pairs of (non-unitarily equivalent) *-homomorphisms with domain C(\mathbbT).

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