CONSTRUCTING ULTRAWEAKLY CONTINUOUS FUNCTIONALS ON B(H)

摘要: In this paper we give a constructive characterisation of ultraweakly continuous linear functionals on the space bounded operators separable Hilbert space. Let H be complex space, with orthonormal basis (en)∞n=1, and B(H) set . The weak operator norm associated (en) is defined by ‖T ‖w ≡ ∞ ∑ j,k=1 2−j−k 〈Tej, ek〉 Weak norms different bases rise to equivalent metrics unit ball B1(H) {T ∈ B (H) : ∀x (‖Tx‖ ≤ ‖x‖)} Moreover, totally respect norm, but completeness that an essentially nonconstructive property; see [2]. discuss, within Bishop’s mathematics [1], those are uniformly some, therefore each, norm. Classically, these precisely ultraweak topology [10]; for reason, shall refer them as B(H). Since bounded, functional f normable, in sense its ‖f‖ sup {|f(x)| x H, ‖x‖ 1} , exists ([1], Ch. 4, (4.3)). classical usually proved using Riesz Representation Theorem HahnBanach (see [7]); unfortunately, order apply each theorems constructively need additional hypotheses about computability certain suprema infima cannot verified present case. Another approach, Received editors September 1, 1995 and, revised form, April 7, 1997. 1991 Mathematics Subject Classification. Primary 46S30. c ©1998 American Mathematical Society 3347 License or copyright restrictions may redistribution; http://www.ams.org/journal-terms-of-use 3348 D. S. BRIDGES AND N. F. DUDLEY WARD taken Kadison Ringrose ([11], Section 7.1), more general context von Neumann algebra theory requires comparison projections; presently developed, latter depends applications Zorn’s lemma. A third proof, which similar spirit ours, found [12] uses version spectral decomposition compact selfadjoint (cf. [4]). Thus there significant obstacles overcome obtaining desired characterisation. show how can surmounted, assume familiarity with, access to, Chapters 4 7 [1]. addition, will following background definitions facts, proofs either [3] standard references such [13], [10], [11]. A(H) elements have adjoints. An element positive if it 〈Ax, x〉 ≥ 0 ; then write 0. If A(H), A∗A 0; square root written |A|. U partial isometry projection P called initial ‖UPx‖ = U(I − )x only U∗ U∗U projection, case UU∗. A(H). We say Hilbert-Schmidt if∑∞n=1 ‖Aen‖ converges, sum series independent ‖A‖2 ( n=1 ‖Aen‖). operator, so A∗; also > bound B, AB BA operators, ‖AB‖2 ‖A‖2, ‖BA‖2 trace class ‖A‖1 〈|A| en, en〉 converges. case, ∑∞ ∥∥∥|A|1/2 en∥∥∥2, |A| (en); moreover, operator. Banach class, A, Tr (A) 〈Aen, (en). Tr(AB) Tr(BA). Now, argument used all does not actually adjoint, require adjoint. It follows for, compact, 1The proposition every l2 has adjoint nonconstructive; Brouwerian Example 3 [9]. CONSTRUCTING ULTRAWEAKLY CONTINUOUS FUNCTIONALS ON 3349 BA, approximable finite-rank hence fA fA(B) Tr(BA) B1 As normable; fact, ‖fA‖ Our first result was (Theorem 1.1). Proposition 1. Approximate polar decomposition: e 0, both A−U U∗A. For decompositions related matters, [6]. Lemma let Then Usuch ‖A− |A|‖1 find V t |(A− |A|)| ∗(A− |A|). q p n=p+1 〈|(A− 〈(|(A− |A|))