James space on general trees (Q1109287)

The paper contains the main results of the author's Ph. D. dissertation. It provides a unified and systematic treatment of the various constructions derived from the by now classical James space. Let T be a tree, i.e. a partially ordered set such that all segments \(\{y| y\leq x\}\) are well-ordered; it will also be assumed that T has a least element 0 and satisfies a certain completeness condition. J(T) is defined to be the collection of all scalar-valued mappings f on T with \(f(0)=0\) which are order-continuous and for which \(\| f\|^ 2:=\sup \sum | f(b_ i)-f(a_ i)|^ 2\) is finite (the supremum runs over the finite families \(a_ i\leq b_ i\), \(i=1,...,n\) such that the segments \(]a_ i,b_ i]\) are disjoint). It is not possible to survey the many results treated in this long paper. They concern e.g. projections on \(J(T)\), transfinite bases, RNP and approximation properties for \(J(T)\) and its (second) duals. Also all duals of J(T) are calculated, and it turns out that \(J(T)^{**}\) is a space of the same type, i.e. it has the form \(J(\tilde T)\) for a suitable tree \(\tilde T.\)