Endomorphisms of B(H

The unital endomorphisms of B(H) of (Powers) index n are classified by certain U(n)-orbits in the set of non-degenerate representations of the Cuntz algebra On on H. Using this, the corresponding conjugacy classes are identified, and a set of labels is given. This set of labels is P/ ∼ where P is a set of pure states on the UHFalgebra Mn∞ , and ∼ is a non-smooth equivalence on P . Several subsets of P , giving concrete examples of non-conjugate shifts, are worked out in detail, including sets of product states, and a set of nearest neighbor states. 0. Introduction Recently the study of endomorphisms of von Neumann algebras has received increased attention, both in connection with the Jones index for subfactors and its applications [Jon], and also in connection with duality for compact groups [Wor] and super-selection sectors in algebraic quantum field theory. Two other articles (by W. Arveson and by R. Powers) in these proceedings deal with semigroups of endomorphisms of the type I∞factor. Here we restrict to the case of single endomorphisms of B(H). Potentially it is expected that the theory for B(H) may possibly be extended or modified to apply also to other factors, but so far only a few relatively isolated results (although still some very important ones) are known for endomorphisms of factors other than B(H). We report here on recent and new developments in the study of End(B(H)). The methods used draw among other things on seminal ideas of von Neumann, and also on ideas of Powers from his pioneering work on the states on the CAR (canonical anticommutation relation)-algebra, and, more generally, states on the UHF (uniformly hyperfinite) Calgebras. The work on End(M) for the case when M is a von Neumann factor of type II1 (especially the hyperfinite case) is ongoing. It will not be treated here, but we refer to [Pow2], [Po-Pr], [EW], [Cho], and [ENWY]. 1. Main Results Let B(H) be the C-algebra of bounded linear operators on the separable, infinite dimensional Hilbert space H. If α : B(H) → B(H) is a unital endomorphism, we say that α is ergodic if {X ∈ B(H) | α(X) = X} = C1, and that α is a shift if 1991 Mathematics Subject Classification. 46L10, 46L50, 47A58, 47C15, 81S99.