tailieunhanh - Báo cáo toán học: "A proof of a theorem on trace representation of strongly positive linear functionals on $OP*-algebras$ "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Một bằng chứng của một định lý về dấu vết của đại diện mạnh mẽ tích cực functionals tuyến tính trên $ OP *- đại số $. | Copyright by INCREST 1979 J. OPERATOR THEORY 2 1979 39 47 A PROOF OF A THEOREM ON TRACE REPRESENTATION OF STRONGLY POSITIVE LINEAR FUNCTIONALS ON Ớ7N-ALGEBRAS KONRAD SCHMŨDGEN 1. INTRODUCTION In 4 the following theorem was shown. Theorem I. Let 2 be a dense linear subspace of a Hilbert space. Suppose that 2 is a Frechet space. The following are equivalent 1 2 f is a Montel space. 2 For any Op -algebra -sd on 2 with each strongly positive linear functional f on sd is a trace functional . f is of the form f a Tr ta aesd where zeSj . The proof given in 4 for the main part 1 2 of the theorem relied on a method developed by Sherman 6 Because Sherman s proof is very long it is desirable to make it simpler. The purpose of the present note is to give another proof of the above-noted result 1 2 which will be stated separately as Theorem II. Let sd be an Op -algebra on 2 and let f be a strongly positive linear functional on sd. Suppose that 2 d is a Frechet-Montel space. Then there exists an operator t e G1 2 such that f a Tr ta for all a Ejd. Our proof of Theorem II is based on topological arguments. It makes use of Theorem more precisely of Proposition in 4 Let US recall some part of Theorem from 4 in a convenient formulation for later use. Proposition. Let sd bean Op -algebra on 2 such that 2 f isa Frechet-Montel space. Let f be a T s-continuous linear functional on sd. Then there is an operator t e Sf2 so that f d Tr ta for all aesd. The present approach to Theorem II is shorter than Sherman s original proof and itj gives a more general result. Some arguments used here independently appear in 1 . In the case of Frechet-Montel domains which have an unconditional basis another proof of Theorem II was also given in 5 40 KONRAD SCHMŨDGEN 2. DEFINITIONS AND NOTATIONS We collect the definitions we use in what follows for more details about Ợp -al-gebras we refer to 2 . Let 3 be a dense linear subspace of a Hilbert space and let L a 6 End 3 a3 c a 3 s 3 . .