Theresa K.-Y. Dodds

Noncommutative Köthe duality

Using techniques drawn from the classical theory of rearrangement invariant Banach function spaces we develop a duality theory in the sense of Kothe for symmetric Banach spaces of measurable operators affiliated with a semifinite von Neumann algebra equipped with a distinguished trace. A principal result of the paper is the identification of the Kothe dual of a given Banach space of measurable operators in terms of normality

Fully symmetric operator spaces

It is shown that certain interpolation theorems for non-commutative symmetric operator spaces can be deduced from their commutative versions. A principal tool is a refinement of the notion of Schmidt decomposition of a measurable operator affiliated with a given semi-finite von Neumann algebra.

A uniform Kadec-Klee property for symmetric operator spaces

We show that if a rearrangement invariant Banach function space E on the positive semi-axis satisfies a non-trivial lower q -estimate with constant 1 then the corresponding space E (M) of τ-measurable operators, affiliated with an arbitrary semi-finite von Neumann algebra M equipped with a distinguished faithful, normal, semi-finite trace τ, has the uniform Kadec-Klee property for the topology of local convergence in measure. In particular, the Lorentz function spaces L q, p and the Lorentz-Schatten classes C g, p have the UKK property for convergence locally in measure and for the weak-operator topology, respectively. As a partial converse, we show that if E has the UKK property with respect to local convergence in measure then E must satisfy some non-trivial lower q -estimate. We also prove a uniform Kadec-Klee result for local convergence in any Banach lattice satisfying a lower q -estimate.

On a Submajorization Inequality of T. Ando

A submajorization inequality of T.Ando for operator monotone functions is extended to the setting of measurable operators affiliated with a semi-finite von Neumann algebra. The general form yields certain norm inequalities for the absolute value in symmetric operator spaces which were previously known in the setting of trace ideals.

Weakly compact subsets of symmetric operator spaces

Under natural conditions it is shown that the rearrangement invariant hull of a weakly compact subset of a properly symmetric Banach space of measurable operators affiliated with a semi-finite von Neumann algebra is again relatively weakly compact.

SOME ASPECTS OF THE THEORY OF SYMMETRIC OPERATOR SPACES

Abstract We discuss several aspects of the theory of symmetric Banach spaces of measurable operators, including their construction and certain topological and geometric properties. Particular emphasis is given to the role played by rearrangement inequalities.

Lipschitz continuity of the absolute value in preduals of semifinite factors

We prove a weak-type estimate for the absolute value mapping in the preduals of semifinite factors which extends an earlier result of Kosaki for the trace class.

Local uniform convexity and Kadec-Klee type properties in K-interpolation spaces I: General Theory

We present a systematic study of the interpolation of local uniform convexity and Kadec-Klee type properties in K-interpolation spaces. Using properties of the K-functional of J.Peetre, our approach is based on a detailed analysis of properties of a Banach couple and properties of a K-interpolation functional which guarantee that a given K-interpolation space is locally uniformly convex, or has a Kadec-Klee property. A central motivation for our study lies in the observation that classical renorming theorems of Kadec and of Davis, Ghoussoub and Lindenstrauss have an interpolation nature. As a partiular by-product of our study, we show that the theorem of Kadec itself, that each separable Banach space admits an equivalent locally uniformly convex norm, follows directly from our approach.

On a singular value inequality of Ky Fan and Hoffman

It is shown that the identity operator is a best unitary approximant to any positive measurable operator affiliated with a semifinite von Neumann algebra equipped with a distinguished faithful normal semifinite trace.