Unitary representations of the groups of measurable and continuous functions with values in the circle

TL;DR

This paper classifies unitary representations of groups of measurable and continuous circle-valued functions, expanding understanding beyond locally compact groups using structure and factorization theorems.

Contribution

It provides the first classification of unitary representations for these non-locally compact groups, utilizing new structure and factorization results.

Findings

Classification of unitary representations for measurable circle-valued functions

Classification of unitary representations for continuous circle-valued functions on zero-dimensional spaces

Use of structure and factorization theorems in proofs

Abstract

We give a classification of unitary representations of certain Polish, not necessarily locally compact, groups: the groups of all measurable functions with values in the circle and the groups of all continuous functions on compact, second countable, zero-dimensional spaces with values in the circle. In the proofs of our classification results, certain structure theorems and factorization theorems for linear operators are used.