Representation Theory and Numerical AF-invariants: The representations and centralizers of certain states on O_d
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.
Keywords
Cite
@article{arxiv.math/9907036,
title = {Representation Theory and Numerical AF-invariants: The representations and centralizers of certain states on O_d},
author = {Ola Bratteli and Palle E. T. Jorgensen and Vasyl Ostrovskyi},
journal= {arXiv preprint arXiv:math/9907036},
year = {2007}
}
Comments
xiv+178 pages, AMS-LaTeX, wncyr font; changes in versions 2 and 3 only in Introduction, except minor typos; more sources in Bibliography (+17 in v2, +3 in v3); accepted for publication in Mem. Amer. Math. Soc