The spectral form of Koopman representations of the group of measurable functions with values in the circle

TL;DR

This paper calculates the spectral form of Koopman representations for a natural boolean action of the group of measurable functions into the circle, confirming earlier constraints on their spectral forms.

Contribution

It provides an explicit computation of the spectral form for Koopman representations of the group of measurable circle-valued functions, validating previous theoretical constraints.

Findings

Spectral form of Koopman representation explicitly computed.

Confirms sharpness of previously established spectral constraints.

Enhances understanding of group actions on measure spaces.

Abstract

We compute the spectral form of the Koopman representation induced by a natural boolean action of $L^0(\lambda, {\mathbb T})$ identified earlier by the authors. Our computation establishes the sharpness of the constraints on spectral forms of Koopman representations of $L^0(\lambda, {\mathbb T})$ previously found by the second author.