Endomorphisms of B(H) .2. Finely correlated states on O-n

We identify sets of conjugacy classes of ergodic endomorphisms of B(H) where H is a fixed separable Hilbert space. They correspond to certain equivalence classes of pure states on the Cuntz algebras O-n where n is the Powers index. These states, called finitely correlated states, and strongly asymptotically shift invariant states, are defined and characterized. The subsets of these stares defining shifts will in general be identified in a later work, but here an interesting cross section for the conjugacy classes of shifts called diagonalizable shifts is introduced and studied. (C) 1997 Academic Press.