Lattice copies of c₀ and in spaces of integrable functions for a vector measure
S. Okada ; W. J. Ricker ; E. A. Sánchez Pérez
GDML_Books, (2014), p.

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)L¹w(m) (which always holds) is proper, then L¹(m) and L¹w(m) contain lattice-isomorphic copies of the complex Banach lattices c₀ and , 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: f ↦ ∫ fdm, for f ∈ L¹(m), 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 1 ≤ 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 . 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.

EUDML-ID : urn:eudml:doc:286055
@book{bwmeta1.element.bwnjournal-rm-doi-10_4064-dm500-0-1,
     author = {S. Okada and W. J. Ricker and E. A. S\'anchez P\'erez},
     title = {Lattice copies of c0 and $l^{[?]}$ in spaces of integrable functions for a vector measure},
     series = {GDML\_Books},
     year = {2014},
     zbl = {1303.28016},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-rm-doi-10_4064-dm500-0-1}
}
S. Okada; W. J. Ricker; E. A. Sánchez Pérez. Lattice copies of c₀ and $ℓ^{∞}$ in spaces of integrable functions for a vector measure. GDML_Books (2014),  http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-rm-doi-10_4064-dm500-0-1/