• 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.