Mathematics – Operator Algebras
Scientific paper
1999-07-06
Mem. Amer. Math. Soc. 168 (2004), no. 797
Mathematics
Operator Algebras
xiv+178 pages, AMS-LaTeX, wncyr font; changes in versions 2 and 3 only in Introduction, except minor typos; more sources in Bi
Scientific paper
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.
Bratteli Ola
Jorgensen Palle E. T.
Ostrovskyi Vasyl
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