# Classes of Banach lattices with disjointness preserving isometries and applications

- Thursday, 19 October
- Plaza de las Ciencias nº 3.Facultad de CC Matemáticas Facultad de CC Matemáticas, Sala 222
Ponente: Yves Raynaud (Institut de Mathematiques de Jussieu, Paris).

If

*X*is a Banach lattice (real or complex), we call \sign change operator" a linear operator*T*:*X ! X*that preserves the modulus, i. e.*jTxj*=*jxj*for every*x 2 X*. We call \sublattice of*X*up to a sign change" any image of a closed vector sublattice by a sign change operator on*X*. We introduce the \Lacey function"*bX*:*X _ X ! X*which is invariant under lattice homomorphisms, sign-change operators and more generally under disjointness preserving bounded operators. We characterize sublattices up to a sign change in an order continuous Banach lattice*X*as the closed linear subspaces of*X*that are*bX*-invariant. As an application we prove that if a Banach space*E*has an ultrapower that is linearly isometric to an order continuous Banach lattice*L*such that every linear isometry from*L*in any of its ultrapowers preserves disjointness, then*E*is linearly isometric to a sublattice of*L*. This result gives a tool for obtaining new classes of axiomatizable Banach spaces. We give various examples of classes of Banach lattices such that any linear isometry from a meer of the class into another one preserves disjointness.-