Journal of Operator Theory

Volume 65, Issue 2, Spring 2011 pp. 451-470.

Lattice isomorphisms between spaces of integrable functions with respect to vector measures

Authors: A. Fernandez (1), F. Mayoral (2), F. Naranjo (3), and E.A. Sanchez-Perez (4)
Author institution: (1) Dpto. Matematica Aplicada II, Escuela Tecnica Superior de Ingenieros, Camino de los Descubrimientos, s/n, 41092-Sevilla, Spain
(2) Dpto. Matematica Aplicada II, Escuela Tecnica Superior de Ingenieros, Camino de los Descubrimientos, s/n, 41092-Sevilla, Spain
(3) Dpto. Matematica Aplicada II, Escuela Tecnica Superior de Ingenieros, Camino de los Descubrimientos, s/n, 41092-Sevilla, Spain
(4) Instituto Universitario de Matematica Pura y Aplicada (I.U.M.P.A.), Universidad Politecnica de Valencia, Camino de Vera, s/n, 46022-Valencia, Spain

Summary: In this paper we study the relation between different spaces of vector measures $(\Omega _1 ,\Sigma _1 ,m_1)$ and $(\Omega _2,\Sigma _2 ,m_2);$ where $(\Omega _1 ,\Sigma _1)$ and $(\Omega _2 ,\Sigma _2)$ are measurable spaces and $m_1$ and $m_2$ are countably additive vector measures taking values in real Banach spaces $X$ and $Y,$ respectively, when the corresponding spaces of integrable functions $L^1 (m_1 )$ and $L^1 (m_2 )$ are lattice isomorphic. As a consequence, we give a description of the lattice isomorphisms between spaces of integrable functions with respect to a vector measure.

Keywords: Vector measure, integrable function, lattice isomorphism, multiplication operator, composition operator, Boolean algebra

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