Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Essentially Commuting with a Unitary

Jui-Hui Chung, Jacob Shapiro

TL;DR

This paper studies operators that essentially commute with a fixed unitary $R$ with spectrum on the unit circle, introducing the $R$-local algebra $\\mathcal{L}_R$ and focusing on the topology of $R$-local unitaries and projections. It establishes that the space of $R$-local unitaries $\\mathcal{U}(\\mathcal{L}_R)$ is path-connected and classifies path components of $R$-non-trivial projections via an index map into $\\mathbb{Z}$, with a bijection on $\\pi_0$ and a maximality property for the nt component. The approach combines K-theory of the $R$-local algebra (noting $K_0(\\mathcal{L}_R)\\cong \\mathbb{Z}$ and $K_1(\\mathcal{L}_R)\\cong 0$) with a concrete Laughlin-flux representation on $\\ell^2(\\mathbb{Z}^2)$ to relate operator locality to topological invariants. These results contribute a robust topological classification relevant to mathematical physics, particularly in contexts related to topological insulators and bulk-edge phenomena, and connect to the theory of $K_1$-injectivity for the Paschke dual algebra.

Abstract

Let $R$ be a unitary operator whose spectrum is the circle. We show that the set of unitaries $U$ which essentially commute with $R$ (i.e., $[U,R]\equiv UR-RU$ is compact) is path-connected. Moreover, we also calculate the set of path-connected components of the orthogonal projections which essentially commute with $R$ and obey a non-triviality condition, and prove it is bijective with $\mathbb{Z}$.

Essentially Commuting with a Unitary

TL;DR

This paper studies operators that essentially commute with a fixed unitary with spectrum on the unit circle, introducing the -local algebra and focusing on the topology of -local unitaries and projections. It establishes that the space of -local unitaries is path-connected and classifies path components of -non-trivial projections via an index map into , with a bijection on and a maximality property for the nt component. The approach combines K-theory of the -local algebra (noting and ) with a concrete Laughlin-flux representation on to relate operator locality to topological invariants. These results contribute a robust topological classification relevant to mathematical physics, particularly in contexts related to topological insulators and bulk-edge phenomena, and connect to the theory of -injectivity for the Paschke dual algebra.

Abstract

Let be a unitary operator whose spectrum is the circle. We show that the set of unitaries which essentially commute with (i.e., is compact) is path-connected. Moreover, we also calculate the set of path-connected components of the orthogonal projections which essentially commute with and obey a non-triviality condition, and prove it is bijective with .
Paper Structure (10 sections, 17 theorems, 94 equations)

This paper contains 10 sections, 17 theorems, 94 equations.

Table of Contents

  1. Introduction and the result
  2. Strategy for the proof
  3. Working with a concrete representation of $R$
  4. Notation
  5. Details of the proofs
  6. Local unitaries
  7. Local projections
  8. Locality and boundedness
  9. The K-theory of local algebra
  10. Counter-examples

Key Result

Theorem 1.1

The space$\,$$\mathcal{U}(\mathcal{L}_R)$ of $R$-local unitaries is path-connected:

Theorems & Definitions (39)

  • Theorem 1.1: Classification of $R$-local unitaries
  • Definition 1.2
  • Remark 1.3
  • Theorem 1.4: Classification of $R$-non-trivial orthogonal projections
  • Remark 1.5
  • Definition 1.6: Projections onto subsets of $\mathbb{Z}^2$ lattice
  • Definition 1.7: Angles of rational slopes
  • Proposition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • ...and 29 more