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Uniqueness of the zeta transformation in operator K-theory

Mikkel Munkholm

TL;DR

The work develops an abstract, unique characterization of the zeta^n natural transformations linking $K_0(-; \mathbb{Z}/n\mathbb{Z})$ to the Hausdorffized unitary algebraic $K_1$ data in the trace-aware invariant for classifiable C*-algebras. It formulates and extends the Thomsen sequence to non-unital algebras, and presents an explicit determinant-based model for $\zeta^n$, proving both existence and a strong uniqueness property independent of the chosen model for $K_*(\cdot; \mathbb{Z}/n\mathbb{Z})$. The results clarify obstructions to lifting invariant morphisms in Elliott-style classification by isolating a canonical natural transformation that encodes trace–K_0 compatibility. Together, these contributions refine the invariant structure used to classify morphisms between simple nuclear C*-algebras supporting $\mathcal{Z}$-absorption and UCT.

Abstract

The classification of homomorphisms between certain unital simple nuclear C*-algebras lead to the discovery of a natural transformation as part of the classifying invariant. We develop a uniqueness result and an abstract characterization of said transformation.

Uniqueness of the zeta transformation in operator K-theory

TL;DR

The work develops an abstract, unique characterization of the zeta^n natural transformations linking to the Hausdorffized unitary algebraic data in the trace-aware invariant for classifiable C*-algebras. It formulates and extends the Thomsen sequence to non-unital algebras, and presents an explicit determinant-based model for , proving both existence and a strong uniqueness property independent of the chosen model for . The results clarify obstructions to lifting invariant morphisms in Elliott-style classification by isolating a canonical natural transformation that encodes trace–K_0 compatibility. Together, these contributions refine the invariant structure used to classify morphisms between simple nuclear C*-algebras supporting -absorption and UCT.

Abstract

The classification of homomorphisms between certain unital simple nuclear C*-algebras lead to the discovery of a natural transformation as part of the classifying invariant. We develop a uniqueness result and an abstract characterization of said transformation.

Paper Structure

This paper contains 6 sections, 6 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. K-theory with $\mathbb{Z}/n\mathbb{Z}$-coefficients
  3. Traces and unitary algebraic $K_1$
  4. One model of $\zeta^n$
  5. Non-unital case
  6. The Main Theorem

Key Result

Theorem 1.1

For each $n\geq 2$, there exists a unique natural transformation $\zeta^n \colon K_0(\, \cdot \, ; \mathbb{Z}/n\mathbb{Z}) \rightarrow \overline{K}^\text{alg}_1$ such that $\mathrlap{\!\not{\space}}\mathrm{a} \circ \zeta^n = \nu^n_0$.

Theorems & Definitions (10)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • Corollary 4.3