Nguyen To Nhu’s research while affiliated with Louisiana State University in Shreveport and other places

http://www.tulips.tsukuba.ac.jp/mylimedio/dl/page.do?issueid=549172&tocid=100080763&page=413-442

A rigid space is a topological vector space whose endomorphisms are all simply scalar multiples of the identity map. This is in sharp contrast to the behavior of operators on l(2), and so rigid spaces are, from the viewpoint of functional analysis, fundamentally different from Hilbert space. Nevertheless, we show in this paper that a rigid space can be constructed which is topologically homeomorphic to Hilbert space. We do this by demonstrating that the first complete rigid space can be modified slightly to be an AR-space (absolute retract), and thus by a theorem of Dobrowolski and Torunczyk is homeomorphic to l(2).

We introduce the notion of the finite dimensional approximation property (the FDAP) and prove that if a subset X of a linear metric space has the FDAP, then every non-empty convex subset of X is an AR. As an application we show that every needle point space X contains a dense linear subspace E with the following properties: (i) E contains a non-empty compact convex set with no extreme points; (ii) all non-empty convex subsets of E are AR.

A compact convex set X in a linear metric space is weakly admissible if for every ε > 0 there exist compact convex subsets X1,…,Xn of X with X = conv(X1 ∪ … ∪ Xn) and continuous maps from Xi into finite dimensional subsets Ei, i = 1, …, n, of X such that for every xiϵXi, and i = 1, …, n.Theorem: Any weakly admissible compact convex set has the fixed point property.Question: Is every weakly admissible compact convex set an AR?

We introduce the notion of the locally convex approximation property (the LCAP) for convex sets in linear metric spaces. The LCAP is an extension of the notion of admissibility of Klee. We prove that any convex set with the LCAP is an AR.

We introduce the notion of the locally convex approximation property (the LCAP) for convex sets in linear metric spaces. The LCAP is an extension of the notion of admissibility of Klee. We prove that any convex set with the LCAP is an AR.