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# 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

#### 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.

Kato was supported by JSPS KAKENHI 17K18725 and 17H06461.

Kishimoto was supported by JSPS KAKENHI 17K05248 and 19K03473.

Tsutaya was supported by JSPS KAKENHI 19K14535.

Received 20 December 2020

Received revised 7 July 2022

Accepted 23 July 2022

Published 2 May 2023