By Constantin, Douglas, Nagy, Voiculescu
Read Online or Download Topics in Modern Operator Theory: 5.Intern.Conference Operator Theory,Timisoara,Herculana,1980 (Operator Theory: Advances and Applications) PDF
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Extra resources for Topics in Modern Operator Theory: 5.Intern.Conference Operator Theory,Timisoara,Herculana,1980 (Operator Theory: Advances and Applications)
Example text
THE CONJECTURE Throughout H shall denote a complex separable infinite-dimensional Hilbert space and 8(H) the bounded linear operators on H. It has often been observed that 8(H), T(H) (the trace class operators) and K(H) are the noncommutative analogues of fm, C1 and c0. In particular 8(H)=T(H)*=K(H)**. We should therefore expect that S(8(H)) and POW)) are the "noncommutative" analogues of S (C°°) and P (f') . Now t'=-C(BIN) where BN denotes the Stone-Cech compactification of the integers. This correspondence arises as follows.
Select self-adjoint operators D andiB with D in D such that IIA-[D,BI1II PROPOSITION 3. The ze,t C i s with A n>(-1/n)1 a prtopen cone in A and A+tC. PROOF. It is trivial that AQC and that C is a cone in A. Suppose aeC(1-C. For each n select An and Bn in B(H)s such that An,Bn_-(1/n)1 and u(An)=u(-Bn)=a. Fix AeB(H)s with u(A)=a. I151/n If (Rk) denotes the refinement of these D-partitions, then ANDERSON 36 (-2/n)RkSRkAnRk+Rk(A-An)Rk=RkARk= =Rk(A+Bn)Rk-RkBnRkS(2/n)Rk so that again by Lemma 1, Let (s) denote the order in A induced by the proper cone C. Note that if a5b in A, then since A+rC, a-b and therefore e is an order unit for (A,3).