TAILIEUCHUNG - Báo cáo toán học: "Subalgebras of reflexive algebras-Erratum "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Subalgebras của đại số lỗi thuộc về phản. | Copyright by INCREST 1986- J. operator theory 15 1986 203-204 SUBALGEBRAS OF REFLEXIVE ALGEBRAS ERRATUM D. w. HADWIN and E. A. NORDGREN The proof of Proposition 4 of 1 does not establish the asserted result. What the proof demonstrates is the following Proposition 1. Ijfsf .i e 1 is a collection of ơ-weakly closed unital algebras and the direct sum of the sJi has property Da then there is an r ỷ ỉ such that ữíị has property Dơ r for all but finitely many i in I. Corollary 2. If sđỊ has property Dfrffor every i in I and the direct sum of the -sf has property Dơ then there is an rỳ Ì such that sđi has property Dff r for every i in I. Corollary 3. A a-weakly closed unital algebra sd has property Dff r for some r 1 if and only if sd sd . has property Dff. Thus the application of Proposition 4 in Theorem of 1 is valid. However the last sentence of part 5 in Section 6 Questions and Comments should be deleted. We wish to point out that the direct integral version of Proposition 4 is true as long as the ơ-finite measure involved is nonatomic. Theorem 4. Suppose X JI p is a nonatomic a-finite measure space Jx X e X fffi is a measurable family of ơ-weakly closed unital algebras and sd sdx dp x . If sd has property D then there is an r 1 such that sd has property D . Proof. Without loss of generality we may assume p X 1. If E is a measur-r able subset of X then let sd E l sd x dp x .Then sd sd E sdX E- Call an alge- E bra ỖH bad if there is no r 1 such thatổ has property D 0 . Suppose sd has property D but sd is bad. We will show that this leads to a contradiction. Since p is nonatomic and p X 1 there exist disjoint measurable sets E and F such that 204 D. w. HADWIN and E. A. NORDGREN p Z O 1 2. Since c aZ sđF j p it follows that either sdE or sf Ị - is bad. Hence there is a measurable subset If such that .NL is bad and p Ef 1 2. Proceeding inductively we obtain a sequence Eỵ E2- . of measureable sets such that rfEn is bad and 1 2 . Let F E E 1 for n 1 2 . .

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