fragmentability and approximation

in chapter 1, charactrizations of fragmentability, which are obtained by namioka (37), ribarska (45) and kenderov-moors (32), are given. also the connection between fragmentability and its variants and other topics in banach spaces such as analytic space, the radone-nikodym property, differentiability of convex functions, kadec renorming are discussed. in chapter 2, we use game characterization or frgmentability of kenderov-moors to construct a large class of non-fragmentable banach space. in particular, we will show that if a compact hausdorff space x contains a non-trivial converging sequence, then (c(x)/c(x), weak) is fragment by any metric, where x is the gleason extremally disconnected space corresponding to x. pedersen proved that if a is a c -algebra with unit, b a chebyshev c subalgebra of a then either ba, b1 or else a, 2 2 matrices, and b is isomorphic to the algebra of diagonal matrices. we extend his result for jb-algebras, in chapter 3, we will show that if b is a chebyshev subalgebra of a untial jb-algebra a, then either b is a trivial subalgebra of a or a has a representation of the from h , where h is a hilbert space. this connects algebraic and geometric properties of a jb-algebra. a famous problem in approximation theory is whether or not sets having unique farthest point property are singletons. klee (33) proved that any positive answer to this problem in hilbert space implies that all chebyshev sets are convex. the problem is considered in chapter 4 for two special cases, i.e, it is shown that every uniquely remotal subset of an alternative jb-algebra or -sum of banach spaces is a singleton.