Classification and realizations of type III factor representations of Cuntz-Krieger algebras associated with quasi-free states

TL;DR

This paper classifies type III factor representations of Cuntz-Krieger algebras linked to quasi-free states and constructs explicit realizations on Hilbert spaces using free groups, bypassing GNS.

Contribution

It provides a complete classification of these representations and offers concrete realizations without GNS, utilizing free groups and type II_1 factors.

Findings

Complete classification of type III factor representations

Explicit realizations on Hilbert spaces without GNS

Use of free groups and type II_1 factors in constructions

Abstract

We completely classify type III factor representations of Cuntz-Krieger algebras associated with quasi-free states up to unitary equivalence. Furthermore, we realize these representations on concrete Hilbert spaces without using GNS construction. Free groups and their type ${\rm II}_{1}$ factor representations are used in these realizations.