Embeddings ofC(Δ) andL
1[0, 1] in Banach lattices |
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Authors: | Heinrich P Lotz Haskell P Rosenthal |
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Institution: | (1) University of Illinois at Urbana-Champaign, 61801 Urbana, Ill., USA |
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Abstract: | It is proved that ifE is a separable Banach lattice withE′ weakly sequentially complete,F is a Banach space andT:E→F is a bounded linear operator withT′F′ non-separable, then there is a subspaceG ofE, isomorphic toC(Δ), such thatT
G is an isomorphism, whereC(Δ) denotes the space of continuous real valued functions on the Cantor discontinuum. This generalizes an earlier result of
the second-named author. A number of conditions are proved equivalent for a Banach latticeE to contain a subspace order isomorphic toC(Δ). Among them are the following:L
1 is lattice isomorphic to a sublattice ofE′;C(Δ)′ is lattice isomorphic to a sublattice ofE′; E contains an order bounded sequence with no weak Cauchy subsequence;E has a separable closed sublatticeF such thatF′ does not have a weak order unit.
The research of both authors was partially supported by the National Science Foundation, NSF Grant No MPS 71-02839 A04. |
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