tailieunhanh - Báo cáo toán học: "Nest-subalgebras of von Neumann algebras: Commutants modulo compacts and distance estimates "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Yến-subalgebras của von Neumann đại số: modulo thể chuyển đổi dòng máy compact và ước tính khoảng cách. | Copyright by INCREST 1982 J. OPERATOR THEORY 7 1982 279-302 NEST-SUBALGEBRAS OF VON NEUMANN ALGEBRAS COMMUTANTS MODULO COMPACTS AND DISTANCE ESTIMATES FRANK GILFEATHER and DAVID R. LARSON INTRODUCTION In recent years several papers have appeared which focus on the structure of the commutant modulo compacts or essential commutant of certain algebras of operators on Hilbert space. Perhaps the most important of these is the work of B. Johnson and s. Parrott 19 in which it was shown that the essential commutant of a von Neumann algebra which does not contain certain intractable type Uj factors as direct summands decomposes as the sum of the algebraic commutant of the von Neumann algebra and the compact operators. Subsequently answering a question of R. Douglas K. Davidson 11 characterized the essential commutant of the analytic Toeplitz operators as the sum of the compact operators and those Toeplitz operators with symbol in 7 c. More recently E. Christensen and c. Peligrad 10 have shown that the essential commutant of an arbitrary nest algebra decomposes as scalar multiples of the identity plus the compact operators. Since the algebraic commutant of a nest algebra is trivial this result is of the same basic type as 19 . In this paper we present a characterization of the essential commutant of a class of operator algebras which generalizes certain aspects of the work of 19 as well as that of 10 . In addition we obtain results concerning the Arveson distance estimate for an operator to certain operator algebras. To a fixed von Neumann algebra and a complete nest of projections contained therein one associates the algebra ữỉ of all operators in ẩă which leave invariant every projection in N. So sá ăữ n sđX where sđJf denotes the nest algebra of in sđ is then a reflexive operator algebra with invariant subspace projection lattice equal to the reflexive lattice generated by jV together with the projections in the commutant of Si in S H . The algebra sđ is called the nest