If and are Banach lattices such that is separable and has the countable interpolation property, then the space of all continuous regular operators has the Riesz decomposition property. This result is a positive answer to a conjecture posed by A. W. Wickstead.
Let be an odd prime number. We prove that if acts freely on a product of equidimensional lens spaces, then . This settles a special case of a conjecture due to C. Allday. We also find further restrictions on non-abelian -groups acting freely on a product of lens spaces. For actions inducing a trivial action on homology, we reach the following characterization: A -group can act freely on a product of lens spaces with a trivial action on homology if and only if and has the -extension property. The main technique is to study group extensions associated to free actions.
We consider real spaces only.
Definition. An operator between Banach spaces and is called a Hahn-Banach operator if for every isometric embedding of the space into a Banach space there exists a norm-preserving extension of to .
A geometric property of Hahn-Banach operators of finite rank acting between finite-dimensional normed spaces is found. This property is used to characterize pairs of finite-dimensional normed spaces such that there exists a Hahn-Banach operator of rank . The latter result is a generalization of a recent result due to B. L. Chalmers and B. Shekhtman.
We consider a class of compact spaces for which the space of probability Radon measures on has countable tightness in the topology. We show that that class contains those compact zero-dimensional spaces for which is weakly Lindelöf, and, under MA + CH, all compact spaces with having property (C) of Corson.
Let be a -uniformly smooth Banach space possessing a weakly sequentially continuous duality map (e.g., ). Let be a Lipschitzian pseudocontractive selfmapping of a nonempty closed convex and bounded subset of and let be arbitrary. Then the iteration sequence defined by , converges strongly to a fixed point of , provided that and have certain properties. If is a Hilbert space, then converges strongly to the unique fixed point of closest to .
Let be a rank two Chevalley group and be the corresponding Moufang polygon. J. Tits proved that is the universal completion of the amalgam formed by three subgroups of : the stabilizer of a point of , the stabilizer of a line incident with , and the stabilizer of an apartment passing through and . We prove a slightly stronger result, in which the exact structure of is not required. Our result can be used in conjunction with the ``weak -pair" theorem of Delgado and Stellmacher in order to identify subgroups of finite groups generated by minimal parabolics.
A commutative Banach algebra is said to have the property if the following holds: Let be a closed subspace of finite codimension such that, for every , the Gelfand transform has at least distinct zeros in , the maximal ideal space of . Then there exists a subset of of cardinality such that vanishes on , the set of common zeros of . In this paper we show that if is compact and nowhere dense, then , the uniform closure of the space of rational functions with poles off , has the property for all . We also investigate the property for the algebra of real continuous functions on a compact Hausdorff space.
Let be an infinite set, a set of pseudo-metrics on and If is limited (finite) for every and every then, for each we can define a pseudo-metric on by writing st We investigate the conditions under which the topology induced on by has a basis consisting only of standard sets. This investigation produces a theory with a variety of applications in functional analysis. For example, a specialization of some of our general results will yield such classical compactness theorems as Schauder's theorem, Mazur's theorem, and Gelfand-Philips's theorem.
We characterize -parabolicity of a noncompact complete Riemannian manifold in terms of the volume growth of under very weak assumptions on . Some of the results also apply to the setting of metric measure spaces.
1. If and is nilpotent of class at most for any , then the group is nilpotent of -bounded class.
2. If and is nilpotent of class at most for any , then the derived group is nilpotent of -bounded class.
Let and be two Del Pezzo fibrations of degrees , respectively. Assume that and differ by a flop. Then we prove that and give a short list of values of other basic numerical invariants of and .
Let be a positive matrix-valued measure on a locally compact abelian group such that is the identity matrix. We give a necessary and sufficient condition on for the absence of a bounded non-constant matrix-valued function on satisfying the convolution equation . This extends Choquet and Deny's theorem for real-valued functions on .
Let be a compact Hausdorff space and let be a separable Hilbert space. We prove that the group of all order automorphisms of the -algebra is algebraically reflexive.