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Symmetric Functionals and Singular Traces Positivity 2: 47–75, 1998. 47 © 1998 Kluwer Academic Publishers. Printed in the Netherlands. Symmetric Functionals and Singular Traces 1 2 3,† 1, P.G. DODDS , B. DE PAGTER , E.M. SEMENOV and F.A. SUKOCHEV ∗ 1Department of Mathematics and Statistics, The Flinders University of South Australia, GPO Box 2100, Adelaide, SA 5001, Australia; 2Department of Mathematics, Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands; 3Department of Mathematics, Voronezh State University, Universitetskaya pl.1, Voronezh 394693, Russia (Received: 11 November 1997; Accepted: 4 December 1997) Abstract. We study the construction and properties of positive linear functionals on symmetric spaces of measurable functions which are monotone with respect to submajorization. The con- struction of such functionals may be lifted to yield the existence of singular traces on certain non- commutative Marcinkiewicz spaces which generalize the notion of Dixmier trace. Mathematics Subject Classifications (1991): 46E30, 46L50 Key words: symmetric functional, singular trace, Marcinkiewicz space, symmetric operator space 1. Introduction and Preliminaries The purpose of this paper is to study the properties of positive linear functionals on symmetric Banach function spaces which are monotone with respect to subma- jorization, in both the commutative and non-commutative setting. Such functionals are called symmetric (see Definition 2.1 below). This notion finds its roots and motivation in the work of Dixmier [7] on the existence of non-trivial traces on the factor B(H) which are singular in the sense that they vanish on all finite rank operators. Such traces have subsequently found important application in non- commutative geometry and quantum field theory [4]. The methods used by Dixmier in his construction of singular traces are based on ideas familiar in the theory of rearrangement invariant spaces; in fact, his approach can be viewed as first constructing a symmetric functional on a certain Marcinkiewicz sequence space and then “lifting” this functional to the ideal of compact operators whose singular value sequences lie in this space. Our objective therefore is to study the existence and properties of symmetric functionals, first in the setting of symmetric spaces E on the positive half-line (see sections 2,3 below) and then to lift the results (see section 4) to the more general setting of symmetric spaces of measurable operators affiliated with some semi-finite von Neumann algebra. † Research supported by RFFI, Grant 95-01-00135 ∗ Research supported by the Australian Research Council. PIPS No.: 159135 (postkap:mathfam) v.1.15 *159135* post56.tex; 23/04/1998; 10:32; p.1 48 P.G. DODDS ET AL. Since any normal symmetric functional is a scalar multiple of integration with respect to Lebesgue measure (see Proposition 2.6), our principal results concern the existence of singular symmetric functionals. The familiar Lebesgue spaces p 1 1 L 0, ), 1 <p as well as the spaces (L L∞) 0, ), (L L∞) 0, ) fail[ to∞ admit any non-trivial≤∞ symmetric functionals;∩ further[ we∞ show (Theorem+ [ 3.1)∞ that there do not exist any non-trivial symmetric functionals on any Lorentz space or on any Orlicz space on 0, ). On the other hand, for Marcinkiewicz spaces [ ∞ Mψ , existence or otherwise of non-trivial symmetric functionals depends crucially on the asymptotic behaviour of the defining concave function ψ. One of the main results of the paper (Theorem 3.4) characterises those Marcinkiewicz spaces which do admit non-trivial singular symmetric functionals. Moreover, by adapting a con- struction of AA Sedaev [21], we show (Theorem 3.9) that if a Marcinkiewicz space admits at least one non-trivial singular symmetric functional, then it admits an uncountable collection of linearly independent such functionals. Finally, in the last section, we show how to reduce the study of symmetric functionals on spaces of measurable operators to the commutative setting. The present paper complements and extends the results in [1], [12]. There are essential differences in setting and it seems that the present approach via the set- ting of symmetric function spaces permits greater structural insight and clarity. For example, our principal results, together with those of Braverman and Mekler [2] show that the existence of a non-trivial singular symmetric functional on a Marcinkiewicz space Mψ on 0, 1) is in fact equivalent to the existence of sym- [ 1 metric subspaces of Mψ which are not interpolation spaces for the pair (L 0, 1), [ L∞ 0, 1)). This fact is of independent interest, and is not immediately accessible from[ the approach of [12]. Some of the results of this paper were presented by the second author to the North West European Analysis Seminar, Lyon, 1995. The first and second authors wish to express their thanks to Thierry Fack for many stimulating discussions concerning the Dixmier trace. We now gather some of the basic notation and terminology that shall be used throughout the paper. A Banach space (E, E)of real-valued measurable func- tions on the interval 0, ) will be calledk·k a symmetric space if (i) E is an ideal lattice (that is, if y [ E∞,andifxis any measurable function on 0, ) with ∈ [ ∞ 0 x y then x E and x E y E), (ii) E is rearrangement-invariant in≤| the sense|≤| that| if y ∈ E,andifk kxis≤k anyk measurable function on 0, ) with ∈ [ ∞ x∗ y∗,thenx Eand x E y E. Here, x∗ denotes the non-increasing, right-continuous= rearrangement∈ k k of =kx givenk by x∗(t) inf s 0 m( x >s ) t ,t>0. = { ≥ : {| | } ≤ } For basic properties of symmetric spaces and rearrangements, we refer to the mono- graphs [14], [3]. We note that for any symmetric space E, 1 1 L 0, ) L∞ 0, ) E L 0, ) L∞ 0, ), [ ∞ ∩ [ ∞ ⊂ ⊂ [ ∞ + [ ∞ post56.tex; 23/04/1998; 10:32; p.2 SYMMETRIC FUNCTIONALS AND SINGULAR TRACES 49 with continuous embeddings. We will denote by E0 the symmetric space associated with E. The space E0 consists of all measurable functions y for which ∞ y E sup x(t)y(t) dt x E, x E 1 < . k k 0 := | | : ∈ k k ≤ ∞ Z0 If E∗ denotes the Banach dual of E, it is known that E0 E∗ and E0 E∗ if ⊂ = and only if the norm E is order-continuous, i.e. from xn E,xn n 0, it k·k { }⊆ ↓ follows that xn E 0. We note that the norm E on the symmetric space E k k → k·k is order-continuous if and only if E is separable, in which case limt x∗(t) 0 for all x E. →∞ = ∈ 1 If x,y L 0, ) L∞ 0, ), we will say that x is submajorized by y and write x y∈if and[ only∞ + if [ ∞ ≺≺ t t x∗(s)ds y∗(s)ds for all t>0. ≤ Z0 Z0 The symmetric space E on 0, ) will be called fully symmetric if and only if [ ∞ x y, y E, implies x E and x E yE. It is well known (see, for ≺≺ ∈ ∈ k k ≤kk example, [14]) that (E, E)is fully symmetric if and only if E is an interpolation 1 k·k space for the pair (L 0, ), L∞ 0, )), with interpolation constant 1. If the norm on E is order continuous,[ ∞ then E[is fully∞ symmetric. 2. Symmetric Functionals on 0, ). [ ∞ Throughout this section, E will denote a fully symmetric Banach function space on 0, ). [ ∞ DEFINITION 2.1. A linear functional 0 φ E∗ will be called symmetric if and only if 0 f, g E and f g imply that≤ φ(f)∈ φ(g). ≤ ∈ ≺≺ ≤ If E L1 0, ), then the functional φ defined by setting φ(f) ∞ f(t)dt for ⊆ [ ∞ = 0 all f E is clearly symmetric. Recall that a functional 0 φ E∗ is called ∈ ≤ ∈R normal (or order continuous)iffn n 0a.e.inEimplies that φ(fn) n 0. For example, see [25]. ↓ ↓ 1 PROPOSITION 2.2. If 0 <φ E∗is normal and symmetric, then E L 0, ) ∈ ⊆ [ ∞ and there exists 0 <c R such that φ(f) c ∞ f(t)dt for all f E. ∈ = 0 ∈ Proof. If 0 <φ E∗is normal, then there exists 0 <w E0such that ∈ R ∈ φ(f) ∞ w(t)f(t)dt, ([25] Theorem 86.3). It follows easily that there exists = 0 0 <c R such that w(t) c a.e. on 0, ). In fact, if there is no such c,then there exist∈R 0 <α<βand measurable= sets[ ∞A, B 0, )such that 0 < m(A) ⊆[ ∞ = m(B) < and w α on A and w β on B. In particular χ A, χ B E and ∞ ≤ ≥ ∈ χ ∗ χ ∗ and so φ(χA) φ(χB).However, A = B = φ(χA) w(t)dt αm(A) < βm(B) w(t)dt φ(χB), = ≤ ≤ = ZA ZB post56.tex; 23/04/1998; 10:32; p.3 50 P.G. DODDS ET AL. and this is a contradiction. 2 Before presenting some non-trivial examples of symmetric functionals, we first show that symmetric functionals are dilation invariant in the sense of of the fol- lowing proposition. For s>0, the dilation operator Ds is defined by setting Dsf(t) f(t/s)for all t>0 and functions f on 0, ). The linear mapping Ds := [ ∞ is bounded on E; moreover, Ds E max 1,s ([14], Corollary 1 to Theorem II k k ≤ { } 4.5). Note that (Dsf)∗ Dsf∗. = PROPOSITION 2.3. If 0 <φ E∗ is symmetric, then φ(Dsf) sφ(f ) for all f E and all s>0. ∈ = ∈ Proof. Let 0 f E.Forn N , there exist disjointly supported 0 ≤ ∈ ∈ ≤ f ,...,fn Eand such that f ∗ f ∗ for all k 1,...,n.Leth f fn. 1 ∈ k = = := 1 +···+ It is easy to see that h∗(t) f ∗(t/n) for all t>0, that is, h∗ Dnf ∗.
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