tailieunhanh - Báo cáo toán học: "Subalgebras of reflexive algebras "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Subalgebras của đại số phản. | Copyright by INCREST 1982 J. OPERATOR THEORY 7 1982 3 23 SUBALGEBRAS OF REFLEXIVE ALGEBRAS Đ. w. HADWIN and E. A. NORDGREN 1. INTRODUCTION An algebra sđ of operators on a Hilbert space is reflexive if the only operators that leave invariant all of the invariant subspaces of jd are the operators belonging to . In other words if Lat sđ p p p p- and 1 P AP 0 for every A in j j and Alg Latj T Latj LatT then 33 is reflexive precisely when Alg Latj . An operator T is reflexive if the weakly closed unital algebra generated by T is reflexive. In 23 D. Sarason proved that all commutative weakly closed unital algebras of normal or of analytic Toeplitz operators are reflexive. The reflexive operators on a finite-dimensional space were characterized by J. Deddens and p. A. Fillmore 8 and all isometries were shown to be reflexive by Deddens 6 . K. Uchiyama 27 has shown that a contraction with unequal defect indices is reflexive and p. Y. Wu 29 has shown that finite defect cu contractions are reflexive. Recently R. Olin and J. Thomson 19 have shown that all subnormal operators are reflexive. Before Olin and Thomson s work w. Wogen 28 extended Deddens result 6 by showing that all quasinormal operators T and T - T commute are reflexive thereby answering a question of Deddens 7 Moreover Wogen 28 proved that if T is quasinormal then every unital weakly closed subalgebra of J T is reflexive. Call an algebra jaf super-reflexive if every unital weakly closed subalgebra of 33 is reflexive and call an operator T super-reflexive if j T is. Then Sarason s results 23 can be reformulated as the unilateral shift and every commutative von Neumann algebra are super-reflexive. In this paper we study some properties of algebras that in the presence of reflexivity imply super-reflexivity. Our main results show that some of these properties are preserved under direct integrals. Using a result of E. Azoff c. K. Fong 4 D. w. HADW1N and E. A. NOROGREN and F. Gilfeather 3 that says that a direct .