T. S. S. R. K. Rao

Diameter-preserving linear bijections of function spaces

In this paper we give a complete description of diameter-preserving linear bijections on the space of affine continuous functions on a compact convex set whose extreme points are split faces. We also give a description of such maps on function algebras considered on their maximal ideal space. We formulate and prove similar results for spaces of vector-valued functions.

There are no denting points in the unit ball of (

For an infinite compact set K and for any Banach space X we show that the unit ball of the space of X-valued functions that are continuous when X is equipped with the weak topology, has no denting points.

Chebyshev centres and centrable sets

In this paper we characterize real Banach spaces whose duals are isometric to L 1 (μ) spaces (the so-called L 1 -predual spaces) as those spaces in which every finite set is centrable. For a locally compact, non-compact set X and for an L 1 -predual E, we give a complete description of the extreme points and denting points of the dual unit ball of C 0 (X, E), equipped with the diameter norm.

Denting and strongly extreme points in the unit ball of spaces of operators

For 1 ≤p ≤ ∞ we show that there are no denting points in the unit ball of ℓ(lp). This extends a result recently proved by Grząślewicz and Scherwentke whenp = 2 [GS1]. We also show that for any Banach spaceX and for any measure space (Ω, A, μ), the unit ball of ℓ(L1 (μ), X) has denting points iffL1(μ) is finite dimensional and the unit ball ofX has a denting point. We also exhibit other classes of Banach spacesX andY for which the unit ball of ℓ(X, Y) has no denting points. When X* has the extreme point intersection property, we show that all ‘nice’ operators in the unit ball of ℓ(X, Y) are strongly extreme points.

Facial characterizations of complex Lindenstrauss spaces

We characterize complex Banach spaces A whose Banach dual spaces are L 1 (μ) spaces in terms of L-ideals generated by certain extremal subsets of the closed unit ball K of A ∗ . Our treatment covers the case of spaces A containing constant functions and also spaces not containing constants. Separable spaces are characterized in terms of w ∗ -compact sets of extreme points of K, whereas the nonseparable spaces necessitate usage of the w ∗ -closed faces of K. Our results represent natural extensions of known characterizations of Choquet simplexes. We obtain also a characterization of complex Lindenstrauss spaces in terms of boundary annihilating measures, and this leads to a characterization of the closed subalgebras of C c (X) which are complex Lindenstrauss spaces.

On remotality for convex sets in Banach spaces

Abstract We show that every infinite dimensional Banach space has a closed and bounded convex set that is not remotal. This settles a problem raised by Sababheh and Khalil in [M. Sababheh, R. Khalil, Remotality of closed bounded convex sets, Numer. Funct. Anal. Optim. 29 (2008) 1166–1170].

CHARACTERIZATIONS OF SOME CLASSES OF L 1-PREDUALS BY THE ALFSEN-EFFROS STRUCTURE TOPOLOGY

Using the Alfsen-Effros structure topology on the extreme boundary of the dual unit ball of a complex Banach space, we give characterizations ofL1-preduals (i.e., Banach spaces whose duals are isometrically isomorphic toL1 (μ) for a non-negative measure μ) and some of its subclasses viz.G-spaces,Cσ-spaces andc0(Г) spaces.

Remark on a Paper of Sababheh and Khalil

We show that any Banach space X that has a sequence of unit vectors weakly converging to 0 has a closed and bounded convex set that is not remotal. This extends the main result of Sababheh and Khalil, (Numer. Funct. Anal. Optim. 2008; 29:1166–1170).

Nice surjections on spaces of operators

A bounded linear operator is said to be nice if its adjoint preserves extreme points of the dual unit ball. Motivated by a description due to Labuschagne and Mascioni [9] of such maps for the space of compact operators on a Hilbert space, in this article we consider a description of nice surjections onK(X, Y) for Banach spacesX, Y. We give necessary and sufficient conditions when nice surjections are given by composition operators. Our results imply automatic continuity of these maps with respect to other topologies on spaces of operators. We also formulate the corresponding result forL(X, Y) thereby proving an analogue of the result from [9] forLp (1 <p ≠ 2 < ∞) spaces. We also formulate results when nice operators are not of the canonical form, extending and correcting the results from [8].

Remarks on a result of Khalil and Saleh

We give a short proof of a recent result that describes onto isometries of L(X, Y) for certain pairs of Banach spaces X, Y.