Date of Award
Ph.D. in Mathematics
In this thesis, we study completions of Archimedean real vector lattices relative to any nonempty set of continuous positively homogeneous functions defined on Rn. Examples of such completions include square mean closed vector lattices and geometric mean closed vector lattices. These functional completions lead to a vector lattice complexification of any Archimedean real vector lattice. Unlike the vector space complexification of an Archimedean real vector lattice, the vector lattice complexification always results in an Archimedean complex vector lattice. For example, we prove that the vector space complexification of the Fremlin tensor product C(X)⊗C(Y) is not a complex vector lattice when X and Y are uncountable metrizable compact spaces. The vector lattice complexification is employed to construct an Archimedean complex vector lattice tensor product, powers of Archimedean complex vector lattices, and the symmetric (antisymmetric) Archimedean complex vector lattice tensor product. We use tensor products and powers to develop a theory for various multilinear maps between Archimedean complex vector lattices. Finally, we prove the Cauchy-Schwarz Inequality for sesquilinear maps from a complex vector space to various types of Archimedean complex vector lattices.
Schwanke, Christopher Michael, "Complex Vector Lattices: Tensor Products And Multilinear Maps" (2015). Electronic Theses and Dissertations. 1068.