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

Ola Bratteli·Palle E. T. Jorgensen·Vasyl Ostrovskyi
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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_αS_α^* =S_{α_{1}}...S_{α_{k}}S_{α_{k}}^*...S_{α_{1}}^* where α=(α_1...α_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_αS_α^*) =(S_{iα}S_{iα}^*)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.

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BibTeXRIS

Ola Bratteli, Palle E. T. Jorgensen, Vasyl Ostrovskyi. 2003-05-27. Representation Theory and Numerical AF-invariants: The representations and centralizers of certain states on O_d. https://arxiv.org/abs/math/9907036

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