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

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.