tailieunhanh - Báo cáo toán học: "Quasitriangular algebras are "maximal"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Đại số Quasitriangular là "tối đa". | Copyright by INCREST 1983 J. OPERATOR THEORY 10 1983 51-56 QUASITRTANGULAR ALGEBRAS ARE MAXIMAL KENNETH R. DAVIDSON Quasitriangular operators were introduced by Halmos 5 as those operators T for which there was an increasing sequence of finite rank projections p tending tọ I such that lim TP 0. In 2 Arveson considered the quasitriangular algebra QT as the set of operators T which are quasitriangular with respect to pn . He showed that QT P al gổ2 -f- X J where X is the ideal of compact operators. Since that time there has been a lot of interest in these algebras. If JI is an arbitrary subspace lattice let Q alg Ji denote the norm closure of alg Ji In this paper we give an affirmative answer to a conjecture in 4 by showing that if Q alg contains a quasitriangular algebra then Qalg. Z is also quasitriangular or is all of In this sense quasitriangular algebras are maximal among compact perturbations of reflexive algebras. When this problem was raised in 4 an important special case was resolved. In this paper a careful analysis of the lattice structure of. will be used to reduce the problem to the special case. In 4 Q alg JI was defined as alg Jt without norm closure. However we have adopted the closure as being more natural. This has proved to be the case for commutative lattices in 1 . In 4 Q alg JI was automatically closed if J was a commutative AF lattice. All of the results in 4 go through if Q alg Jt is replaced by its closure. The only place where nontrivial differences must be made is in Theorem . We will outline the changes we require later in this paper. All Hilbert spaces in this paper are separable. Subspace lattices are assumed to be complete and closed in the strong operator topology. As in 4 a lattice is called AF if every element is the sup of finite rank elements of the lattice. For commutative lattices it suffices that the identity is the up of finite rank elements. The symbols V and A will denote the lattice operations of sup closed linear span .