Lattice copies of c(0) and l infinity in spaces of integrable functions for a vector measure

Abstract

The spaces L1(m) of all m-integrable (resp. L1w(m) of all scalarly m-integrable) functions for a vector measure m, taking values in a complex locally convex Hausdorff space X (briefly, lcHs), are themselves lcHs for the mean convergence topology. Additionally, L1w(m) is always a complex vector lattice; this is not necessarily so for L1(m). To identify precisely when L1(m) is also a complex vector lattice is one of our central aims. Whenever X is sequentially complete, then this is the case. If, additionally, the inclusion L1(m) in L1w(m) (which always holds) is proper, then L1(m) and L1w(m) contain lattice-isomorphic copies of the complex Banach lattices c0 and `1, respectively. On the other hand, whenever L1(m) contains an isomorphic copy of c0, merely in the lcHs sense, then necessarily L1(m) ( L1w(m). Moreover, the X-valued integration operator Im, then fixes a copy of c0. For X a Banach space, the validity of L1(m) = L1w(m) turns out to be equivalent to Im being weakly completely continuous. A sufficient condition for this is the (q; 1)-concavity of Im for some q . This criterion is fulfilled when Im belongs to various classical operator ideals. Unlike for L1w(m), the space L1(m) can never contain an isomorphic copy of l infinity. A rich supply of examples and counterexamples is presented. The methods involved are a hybrid of vector measure/integration theory, functional analysis, operator theory and complex vector lattices.