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Free and projective Banach lattices

Published online by Cambridge University Press:  30 January 2015

Ben de Pagter
Affiliation:
Delft Institute of Applied Mathematics, Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, PO Box 5031, 2600 GA Delft, The Netherlands, ([email protected])
Anthony W. Wickstead
Affiliation:
Pure Mathematics Research Centre, Queen's University Belfast, Belfast BT7 1NN, Northern Ireland, ([email protected])

Abstract

We define and prove the existence of free Banach lattices in the category of Banach lattices and contractive lattice homomorphisms, and establish some of their fundamental properties. We give much more detailed results about their structure in the case when there are only a finite number of generators, and give several Banach lattice characterizations of the number of generators being, respectively, one, finite or countable. We define a Banach lattice P to be projective if, whenever X is a Banach lattice, J is a closed ideal in X, Q : XX/J is the quotient map, T : PX/J is a linear lattice homomorphism and ε > 0, there exists a linear lattice homomorphism : PX such that T = Q º and ∥∥ ≤ (1 + ε)∥T∥. We establish the connection between projective Banach lattices and free Banach lattices, describe several families of Banach lattices that are projective and prove that some are not.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2015 

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