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

Tsuyoshi Kato; Daisuke Kishimoto; Mitsunobu Tsutaya

doi:10.4310/hha.2023.v25.n1.a20

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arXiv · 2007.06787

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

Tsuyoshi Kato·Daisuke Kishimoto·Mitsunobu Tsutaya

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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.

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BibTeX RIS

Tsuyoshi Kato, Daisuke Kishimoto, Mitsunobu Tsutaya. 2020-07-14. Homotopy type of the space of finite propagation unitary operators on $\mathbb{Z}$. https://doi.org/10.4310/hha.2023.v25.n1.a20

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