Representation Theory and Numerical AF-invariants: The representations and centralizers of certain states on O_d

TL;DR

This paper explores the structure of representations of the Cuntz algebra O_d, focusing on the spectral analysis of its abelian subalgebra D_d, and introduces methods to classify associated AF-algebras arising from these representations.

Contribution

It provides a new spectral approach to express operators in representations of O_d and develops effective methods for classifying related AF-algebras.

Findings

Operators S_i can be expressed via the spectral representation of D_d.

A class of type III representations of O_d is described.

Methods for deciding isomorphism of AF-algebras are developed.

Abstract

Let O_d be the Cuntz algebra on generators S_1,...,S_d, 2 \leq d < \infty, and let D_d \subset O_d be the abelian subalgebra generated by monomials S_\alpha S_\alpha^* =S_{\alpha_{1}}...S_{\alpha_{k}}S_{\alpha_{k}}^*...S_{\alpha_{1}}^* where \alpha=(\alpha_1...\alpha_k) ranges over all multi-indices formed from {1,...,d}. In any representation of O_d, D_d may be simultaneously diagonalized. Using S_i(S_\alpha S_\alpha^*) =(S_{i\alpha}S_{i\alpha}^*)S_i, we show that the operators S_i from a general representation of O_d may be expressed directly in terms of the spectral representation of D_d. We use this in describing a class of type III representations of O_d and corresponding endomorphisms, and the heart of the paper is a description of an associated family of AF-algebras arising as the fixed-point algebras of the associated modular automorphism groups. Chapters 5--18 are devoted to finding effective methods to decide isomorphism and non-isomorphism in this class of AF-algebras.