Endomorphisms of B(h)

Abstract

The unital endomorphisms of B(H) of (Powers) index n are classiied 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 identiied, and a set of labels is given. This set of labels is P= where P is a set of pure states on the UHF-algebra M n 1, 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 eld theory. Two other articles (by W. Arveson and by R. Powers) in these proceedings deal with semigroups of endomorphisms of the type I 1-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 modiied 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 hyperrnite) C-algebras. The work on End(M) for the case when M is a von Neumann factor of type II 1 (especially the hyperrnite 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, innnite dimensional Hilbert space H. If : B(H) ! B(H) is a unital endomorphism, we say that is ergodic if fX 2 B(H) j (X) = Xg = C 1, and that is a shift if

Cite this paper

@inproceedings{BRATTELI1991EndomorphismsOB, title={Endomorphisms of B(h)}, author={OLA BRATTELI and Palle Jorgensen and GEOFFREY L . PRICET}, year={1991} }