1997/09/26 by Neal Fowler, Fowler, Neal, Marcelo Laca +1
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #funct-an #math.OA
paper · pdf · doi:10.48550/arxiv.funct-an/9709004
22 pages, AMS-LaTeX
arxiv created 1997/09/26 · arxiv updated 2009/11/30
Let Fn be the fixed-point algebra of the gauge action of the circle on the Cuntz algebra On. For every pure state ρof Fn and every representation θof C(T) we construct a representation of On, and we use the resulting class of representations to parameterize the space of all states of On which extend ρ. We show that the gauge group acts transitively on the pure extensions of ρand that the action is p-to-1 with p the period of ρunder the usual shift. We then use the above representations of On to construct endomorphisms of B(H) which we classify up to conjugacy in terms of the parameters ρand θ. In particular our construction yields every ergodic endomorphism αwhose tail algebra \bigcapkαk(B(H)) has a minimal projection, and our results classify these ergodic endomorphisms by an equivalence relation on the pure states of Fn. As examples we analyze the ergodic endomorphisms arising from periodic pure product states of Fn, for which we are able to give a geometric complete conjugacy invariant, generalizing results of Stacey, Laca, and Bratteli-Jorgensen-Price on the shifts of Powers.