Constructing Ultraweakly Continuous Functionals on B(h)

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