tailieunhanh - Báo cáo toán học: "Some applications of a technique for constructing reflexive operator algebras "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Một số ứng dụng của một kỹ thuật để xây dựng các đại số nhà điều hành phản. | J. OPERATOR THEORY 227 236 Copyright by INCREST 1985 SOME APPLICATIONS OF A TECHNIQUE FOR CONSTRUCTING REFLEXIVE OPERATOR ALGEBRAS JON KRAUS and DAVID R. LARSON In 4 w. Arveson showed that if sđ is a nest algebra then the distance from an arbitrary operator T in L H to si is given by the formula d T si supfllPJ-TPIIrPe lata . E. c. Lance obtained this result independently 15 . This distance formula has proven to be very useful in investigating problems involving compact perturbations and similarity theory for nests I 2 4 11 12 15 17 . Certain other reflexive algebras have been shown to possess the property that there exists a constant K such that d T J A7sup pj- 7P p 6 lat st for all T in L H 7 8 9 10 13 . Simple examples show that this constant need not be 1. K. Davidson asked in 10 whether every reflexive algebra has such a distance estimate for some constant K. This question was also posed in 13 15 16 . In 5 w. Arveson conjectured that the answer to this question is no. In this paper we verify this conjecture by producing an explicit example of a reflexive algebra that does not satisfy a distance estimate for any K. We also give examples of points of discontinuity of the maps alg and st lat st where the topology for both lattices and algebras is that induced by the Hausdorff metric. We would like to thank w. Arveson for several helpful conversations concerning this paper and in particular for pointing out to US that if y is a reflexive subspace of L H then 7C y VO c J is a reflexive subalgebra of L ỉĩ H . We use this observation to construct reflexive algebras with certain properties by first constructing reflexive subspaces with analogous properties. The research for this paper was done while both authors were visiting the University of California at Berkeley. 228 JON KRAUS and DAVID R. LARSON Ỉ. REFLEXIVE AND HYPERREFLEXIVE SUBSPACES OF In 18 Loginov and Sulman introduced the following notion of a reflexive linear subspace of L H . Definition a .