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Proof of Kitaev determinant trivialization conjecture

Guo Chuan Thiang

Abstract

Using ideas from algebraic $K$-theory, we prove that a simple and naturally applicable criterion of Kitaev suffices to trivialize the Fredholm determinant of a multiplicative commutator.

Proof of Kitaev determinant trivialization conjecture

Abstract

Using ideas from algebraic -theory, we prove that a simple and naturally applicable criterion of Kitaev suffices to trivialize the Fredholm determinant of a multiplicative commutator.

Paper Structure

This paper contains 5 sections, 4 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Proof of determinant trivialization theorem
  3. Discussion
  4. $K$-theory perspective
  5. Application to quantized traces

Key Result

Theorem 1.1

If $U,V\in\mathcal{L}^\times$ satisfy then $\det(UVU^{-1}V^{-1})=1$.

Theorems & Definitions (9)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • proof : Proof of Theorem \ref{['thm:Kitaev.vanishing']}
  • Theorem 3.1
  • proof
  • Remark 3.2