TAILIEUCHUNG - Báo cáo toán học: "Compact operators in the radical of a reflexive operator algebra "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Các nhà khai thác nhỏ gọn trong các gốc tự do của một đại số nhà điều hành phản. | J. OPERATOR THEORY 2 1979 127-129 Copyright by INCREST 1979 COMPACT OPERATORS IN THE RADICAL OF A REFLEXIVE OPERATOR ALGEBRA ALAN HOPENWASSER In the appendix to 2 Fall Arveson and Muhly characterize the compact operators which lie in the radical of a nest algebra. The purpose of this note is to show that this characterization remains valid for any reflexive operator algebra whose invariant subspace lattice is commutative. Furthermore the proof we give is much shorter and simpler than the proof in 2 It should be pointed out that essentially similar results are obtained in 1 for a nest algebra of a maximal nest. Let ỈP be a commutative subspace lattice acting on a separable Hilbert space and let sd sểíỹ ỵ. Let X denote the carrier space of j the set of all lattice homomorphisms of ÍẾ onto the trivial lattice 0 1 . A projection E in J is said to be an interval from 5 if E P Q where P Q ÍẾ and Q p. If p is an element of X such that p P 1 and p 2 0 we say that E is a test interval for p. The family of all test intervals for p has the finite intersection property and is maximal with respect to possession of this property. An interval E is an atom if either E p or E_LP for all p 6 ỈẾ. Each atom E gives rise to an element j9 of the carrier space defined by p P 1 if E p and p P 0 if . Any element of X which arises in this way is said to be atomic the others are non-atomic. If p is atomic the family SF v consists of all intervals which contain the associated atom E. If SF tỊ is non-atomic SFv is a directed set which converges strongly to 0. For each p e X a semi-norm Ap is defined on J by -X pCT infdlETEII l-Ee The set aa Ợ 7 yrí T 0 is a closed two sided ideal in j . The intersection y of all the ideals Xp is always contained in the radical Ổ of 3 . In a variety of cases including all nest algebras it is known that y More information about these concepts and proofs of the various assertions above can be found in 3 4 . Define a mapping Ô from into the commutant of the .

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