Non-commutative subsequence principles

We prove that if is a hyperfinite semi-finite von Neumann algebra equipped with a normal semi-finite trace and (x _n ) _{n = 1} is a bounded sequence in L ¹ ( ) then there exists a subsequence (y _n ) _{n = 1} of (x _n ) _{n = 1} such that for any further subsequence (z _n ) _{n = 1} of (y _n ) _{n = 1} and > 0, there exists a projection p with (1–p)< and thus providing a non-commutative analogue of the classical Komlóss subsequence theorem. We also extend a classical result of Menchoff and Marcienkiewicz on convergence almost everywhere of subseries of orthogonal functions in L ² [0,1] to orthogonal operators in the non-commutative L ² -space associated to the type II ₁ hyperfinite factor. Mathematics Subject Classification (2000):46L51, 46L52, 40A05, 60F15.Supported in part by NSF grant DMS-0096696 and by a Miami University Summer Research Appointment.     

Fixed point properties of semigroups of nonexpansive mappings

We prove that if H is a Hilbert space then the Schatten (trace) class operators on H has the weak^∗ fixed point property for left reversible semigroups. This answered positively a problem raised by A.T.-M. Lau. We also prove that if M is a finite von Neumann algebra then any nonempty bounded convex subset of the non-commutative L¹-space associated to M that is compact for the measure topology has the fixed point property for left reversible semigroups.     

A weak Grothendieck compactness principle

The Grothendieck compactness principle states that every norm compact subset of a Banach space is contained in the closed convex hull of a norm null sequence. In this article, an analogue of the Grothendieck compactness principle is considered when the norm topology of a Banach space is replaced by its weak topology. It is shown that every weakly compact subset of a Banach space is contained in the closed convex hull of a weakly null sequence if and only if the Banach space has the Schur property.     

On complemented versions of James's distortion theorems

Examples are given to show that two natural questions asked in Dowling, Randrianantoanina, and Turett, 1999, about complemented versions of James's distortion theorems have negative answers.

     

On the extension of Hölder maps with values in spaces of continuous functions

We study the isometric extension problem for Hölder maps from subsets of any Banach space intoc ₀ or into a space of continuous functions. For a Banach spaceX, we prove that anyα-Hölder map, with 0<α ≤1, from a subset ofX intoc ₀ can be isometrically extended toX if and only ifX is finite dimensional. For a finite dimensional normed spaceX and for a compact metric spaceK, we prove that the set ofα’s for which allα-Hölder maps from a subset ofX intoC(K) can be extended isometrically is either (0, 1] or (0, 1) and we give examples of both occurrences. We also prove that for any metric spaceX, the above described set ofα’s does not depend onK, but only on finiteness ofK.     

Compact range property and operators on -algebras

We prove that a Banach space has the compact range property (CRP) if and only if, for any given -algebra , every absolutely summing operator from into is compact. Related results for -summing operators () are also discussed as well as operators on non-commutative -spaces and -summing operators.

     

Square function inequalities for non-commutative martingales

We prove a non-commutative version of the weak-type (1,1) boundedness of square functions of martingales. More precisely, we prove that there is an absolute constantK with the following property: ifM is a semifinite von Neumann algebra with a faithful normal traceτ and (M _n ) _n=1 ^∞ is an increasing filtration of von Neumann subalgebras of (M then for any martingalex= _n=1 ^∞ inL ¹(M,τ), adapted to (M _n ) _n=1 ^∞ , there is a decomposition into two sequences (x _n ) _n=1 ^∞ and (z _n ) _n=1 ^∞ withx _n=y _n+z _nfor everyn≥1 and such that . This generalizes a result of Burkholder from classical martingale theory to non-commutative martingales. We also include some applications to martingale Hardy spaces. Supported in part by NSF grant DMS-0096696.     

Fixed point properties in Hardy spaces

We prove that the Hardy space H¹ on the unit disc of C has the weak^∗ fixed point property for left reversible semigroups.