Continuous categories of endomorphisms associated with $G$-kernels

TL;DR

This paper extends the construction of tensor categories of endomorphisms from discrete groups to compact second countable groups, creating a continuous framework linking subfactor theory and group representations.

Contribution

It introduces a novel approach using a unitary tensor functor from $C(G)$-modules to endomorphisms of a type III factor, generalizing previous discrete group methods.

Findings

Constructs a continuous category of endomorphisms for compact groups.

Establishes a functor from $C(G)$-modules to endomorphisms of $M$.

Provides a new perspective on the relationship between subfactors and continuous group representations.

Abstract

We generalize the construction of tensor categories of endomorphisms of a type III factor $M$ associated with a $G$-kernel, from the case of a discrete group $G$ to that of a compact second countable group. Our approach is based on the construction of a unitary tensor functor from a category of $C(G)$-modules to the category of endomorphisms of $M$. This functor maps a $C(G)$-module, realized as the space of square-integrable functions on a measure space, to a continuous family of endomorphisms of $M$. The resulting structure is a continuous category of endomorphisms, which provides a new framework for studying the interplay between subfactor theory and the representation theory of continuous groups.