Endomorphisms of B(H), extensions of pure states, and a class of representations of O_n

Abstract

Let F_n be the fixed-point algebra of the gauge action of the circle on the Cuntz algebra O_n. For every pure state \rho of F_n and every representation \theta of C(T) we construct a representation of O_n, and we use the resulting class of representations to parameterize the space of all states of O_n which extend \rho. We show that the gauge group acts transitively on the pure extensions of \rho and that the action is p-to-1 with p the period of \rho under the usual shift. We then use the above representations of O_n to construct endomorphisms of B(H) which we classify up to conjugacy in terms of the parameters \rho and \theta. In particular our construction yields every ergodic endomorphism \alpha whose tail algebra $\bigcap_k\alpha^k(B(H))$ has a minimal projection, and our results classify these ergodic endomorphisms by an equivalence relation on the pure states of F_n. As examples we analyze the ergodic endomorphisms arising from periodic pure product states of F_n, 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.