Positivity 2: 47–75, 1998.
© 1998 Kluwer Academic Publishers. Printed in the Netherlands.
Symmetric Functionals and Singular Traces
, B. DE PAGTER
, E.M. SEMENOV
and F.A. SUKOCHEV
Department of Mathematics and Statistics, The Flinders University of South Australia, GPO Box
2100, Adelaide, SA 5001, Australia;
Department of Mathematics, Delft University of Technology,
Mekelweg 4, 2628 CD Delft, The Netherlands;
Department 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  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 . 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.
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