TY - JOUR

T1 - Free and projective Banach lattices

AU - de Pagter, Ben

AU - Wickstead, Anthony W.

N1 - Accepted author manuscript

PY - 2015

Y1 - 2015

N2 - 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 : X → X/J is the quotient map, T : P → X/J is a linear lattice homomorphism and ε > 0, there exists a linear lattice homomorphism : P → X 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.

AB - 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 : X → X/J is the quotient map, T : P → X/J is a linear lattice homomorphism and ε > 0, there exists a linear lattice homomorphism : P → X 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.

UR - http://resolver.tudelft.nl/uuid:b6caf6f0-5327-4531-9836-d7846e98df30

U2 - 10.1017/S0308210512001709

DO - 10.1017/S0308210512001709

M3 - Article

VL - 145

SP - 105

EP - 143

JO - Royal Society of Edinburgh. Proceedings. Section A(Mathematics)

JF - Royal Society of Edinburgh. Proceedings. Section A(Mathematics)

SN - 0308-2105

IS - 1

ER -