Continuity of monotone functions with values in Banach lattices
by
Lech Drewnowski
Faculty of Mathematics and Computer Science, A. Mickiewicz University, Poznan, Poland

Let us say that a Banach lattice E has property (l) if every increasing function f [0, 1]® E has at most countably many discontinuities. In 1992, B. Lavric gave a (two page long) construction of an increasing function f [0, 1]® l_¥ without points of continuity. He then combined this with a classical Lozanovskii - Meyer-Nieberg theorem to prove that a s-Dedekind complete Banach lattice contains no lattice copy of l_¥ iff it has property (l). He also proved that every separable Banach lattice has property (l), and asked if a Banach lattice without lattice copies of l_¥ must have property (l).

We first give a short construction of an increasing function f [0, 1]®l_¥ with no points of continuity. Then we extend Lavric's results to the case of nonlocally convex F-normed lattices (which is easy). Next, we answer his question in the negative by proving that the Banach lattice of regular functions on [0, 1] contains no copy of l_¥, and yet lacks property (l). Since this is isomorphically a C(K) space, a natural question arises: If E is a Banach lattice with property (l), for what compact (or locally compact ) spaces K also C(K, E) has property (l) ? We showed so far, for instance, that it is so whenever K is an Eberlein compact (in particular, a metrizable compact space), or an arbitrary product of Eberlein compacts. A similar question is considered for spaces of E-valued Bochner measurable functions. It is proved, e.g., that the spaces L_p(m, E), for any measure m and 1 £ p < ¥, have property (l) whenever E has it.

(T)

Date received: November 30, 1999