Endomorphisms of B(H), extensions of pure states, and a class of representations of O-n

We construct the pure states of O-n that extend a given pure state of the f ixed point algebra F-n of the gauge action, and we show that the gauge grou p acts transitively on these extensions. We apply this to construct and cla ssify the ergodic endomorphisms of B(H) whose tail algebra has a minimal pr ojection. We discuss examples arising from product states of F-n and from t he trace on the Choi subalgebra of O-n.