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

TL;DR

This paper constructs and classifies a broad class of endomorphisms of B(H) using representations of the Cuntz algebra O_n, linking pure states of a fixed-point algebra to ergodic endomorphisms and providing geometric invariants.

Contribution

It introduces a new parametrization of states of O_n extending pure states of F_n and classifies ergodic endomorphisms via these parameters, generalizing previous results.

Findings

Constructed representations of O_n from pure states and circle representations.

Classified ergodic endomorphisms with minimal tail algebra projections.

Provided geometric invariants for shifts arising from periodic pure product states.

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.