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

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