Measurable bundles of Banach lattices
The theory of Banach bundles has proved to have many applications in analysis. For example, continuous and measurable Banach bundles are used for representing various objects in functional analysis. A Banach-Kantorovich space over a Dedekind complete vector lattice is linearly isometric to the space of almost global sections of a suitable continuous Banach bundle over an extremally disconnected compactum. The uniqueness class for this representation was established by \textit{A. E. Gutman} in [Sib. Adv. Math. 3, No.~3, 1--55 (1993; Zbl 0854.46004)]. The next milestone was to specify this representation result if the norming lattice is an ideal function space. In this case, the representing object is a measurable Banach bundle and the uniqueness class is comprised of liftable Banach bundles. This is also due to A. E. Gutman. The objective of the present paper is to outline a portion of Gutman's theory of liftable Banach bundles of Banach lattices and to give applications to the representation of dominable operators.
scientific article; zbMATH DE number 5837808