# Lattice copies of c₀ and ${\ell}^{\infty}$ in spaces of integrable functions for a vector measure

S. Okada; W. J. Ricker; E. A. Sánchez Pérez

- 2014

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topS. Okada, W. J. Ricker, and E. A. Sánchez Pérez. Lattice copies of c₀ and $ℓ^{∞}$ in spaces of integrable functions for a vector measure. 2014. <http://eudml.org/doc/286055>.

@book{S2014,

abstract = {The spaces L¹(m) of all m-integrable (resp. $L¹_\{w\}(m)$ 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, $L¹_\{w\}(m)$ 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 $L¹(m) ⊆ L¹_\{w\}(m)$ (which always holds) is proper, then L¹(m) and $L¹_\{w\}(m)$ contain lattice-isomorphic copies of the complex Banach lattices c₀ and $ℓ^∞$, respectively. On the other hand, whenever L¹(m) contains an isomorphic copy of c₀, merely in the lcHs sense, then necessarily $L¹(m) ⊊ L¹_\{w\}(m)$. 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 $L¹(m) = L¹_\{w\}(m)$ 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 $L¹_\{w\}(m)$, the space L¹(m) can never contain an isomorphic copy of $ℓ^\{∞\}$. 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.},

author = {S. Okada, W. J. Ricker, E. A. Sánchez Pérez},

keywords = {vector measure; space of integrable functions; locally convex space; complex vector lattice; operator ideal vector measure; space of integrable functions; integration operator},

language = {eng},

title = {Lattice copies of c₀ and $ℓ^\{∞\}$ in spaces of integrable functions for a vector measure},

url = {http://eudml.org/doc/286055},

year = {2014},

}

TY - BOOK

AU - S. Okada

AU - W. J. Ricker

AU - E. A. Sánchez Pérez

TI - Lattice copies of c₀ and $ℓ^{∞}$ in spaces of integrable functions for a vector measure

PY - 2014

AB - The spaces L¹(m) of all m-integrable (resp. $L¹_{w}(m)$ 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, $L¹_{w}(m)$ 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 $L¹(m) ⊆ L¹_{w}(m)$ (which always holds) is proper, then L¹(m) and $L¹_{w}(m)$ contain lattice-isomorphic copies of the complex Banach lattices c₀ and $ℓ^∞$, respectively. On the other hand, whenever L¹(m) contains an isomorphic copy of c₀, merely in the lcHs sense, then necessarily $L¹(m) ⊊ L¹_{w}(m)$. 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 $L¹(m) = L¹_{w}(m)$ 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 $L¹_{w}(m)$, the space L¹(m) can never contain an isomorphic copy of $ℓ^{∞}$. 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.

LA - eng

KW - vector measure; space of integrable functions; locally convex space; complex vector lattice; operator ideal vector measure; space of integrable functions; integration operator

UR - http://eudml.org/doc/286055

ER -

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