1999/07/31 by Ola Bratteli, Palle E. T. Jorgensen, Vasyl Ostrovskyi · 1 citation
Mathematics · #math.OA #math.DS #math.FA #math.RT #msc:46L30 #msc:46L55 #msc:46L89 #msc:47A13 #msc:47A67 #msc:47A20 #msc:47D25 #msc:43A65
published as Mem. Amer. Math. Soc. 168 (2004), no. 797 · 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
arxiv created 2003/05/27 · arxiv updated 2009/11/30
Let Od be the Cuntz algebra on generators S1,...,Sd, 2 ≤ d < ∞, and let Dd ⊂ Od be the abelian subalgebra generated by monomials SαSα^* =S_α1...S_αkS_αk^*...S_α1^* where α=(α1...αk) ranges over all multi-indices formed from 1,...,d. In any representation of Od, Dd may be simultaneously diagonalized. Using Si(SαSα^*) =(SiαSiα^*)Si, we show that the operators Si from a general representation of Od may be expressed directly in terms of the spectral representation of Dd. We use this in describing a class of type III representations of Od 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.