Homology, Homotopy and Applications

Volume 25 (2023)

Number 1

Homotopy type of the space of finite propagation unitary operators on $\mathbb{Z}$

Pages: 375 – 400

DOI: https://dx.doi.org/10.4310/HHA.2023.v25.n1.a20

Authors

Tsuyoshi Kato (Department of Mathematics, Kyoto University, Kyoto, Japan)

Daisuke Kishimoto (Faculty of Mathematics, Kyushu University, Fukuoka, Japan)

Mitsunobu Tsutaya (Faculty of Mathematics, Kyushu University, Fukuoka, Japan)

Abstract

The index theory for the space of finite propagation unitary operators was developed by Gross, Nesme, Vogts and Werner from the viewpoint of quantum walks in mathematical physics. In particular, they proved that $\pi_0$ of the space is determined by the index. However, nothing is known about the higher homotopy groups. In this article, we describe the homotopy type of the space of finite propagation unitary operators on the Hilbert space of square summable $\mathbb{C}$-valued $\mathbb{Z}$-sequences, so we can determine its homotopy groups. We also study the space of (end-)periodic finite propagation unitary operators.

Keywords

finite propagation, unitary operator, homotopy group, homotopy type, Grassmannian

2010 Mathematics Subject Classification

Primary 55Q52. Secondary 46L80, 81R10.

Received 20 December 2020

Received revised 7 July 2022

Accepted 23 July 2022

Published 2 May 2023