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https://hdl.handle.net/2142/21373
Description
Title
Matrix-regular orders on operator spaces
Author(s)
Schreiner, Walter James
Issue Date
1995
Doctoral Committee Chair(s)
Berg, I. David
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
Abstract
In this thesis, the concept of the regular (or Riesz) norm on ordered real Banach spaces is generalized to matrix ordered complex operator spaces in a way that respects the matricial structure of the operator space. A norm on an ordered real Banach space E is regular if: (1) $-x \le y \le x$ implies that $\Vert y\Vert \le\Vert x\Vert;$ (2) $\Vert y\Vert < 1$ implies the existence of $x \in E$ such that $\Vert x \Vert < 1$ and $-x \le y \le x.$ A matrix ordered operator space is called matrix regular if, at each matrix level, the restriction of the norm to the self-adjoint elements is a regular norm. In such a space, elements at each matrix level can be written as linear combinations of four positive elements.
After providing the necessary background material on operator spaces, especially with respect to the Haagerup ($\otimes\sp{h}),$ operator projective $(\\otimes),$ and operator injective $(\check\otimes)$ tensor products, the concept of the matrix ordered operator space is made specific in such a way as to be a natural generalization of ordered real and complex Banach spaces. For the case where V is a matrix ordered operator space, a natural cone is defined on the operator space $X\sp* \otimes \sp{h} V \otimes\sp{h}\ X$ so as to make it a matrix ordered operator space. Exploiting the advantages gained by taking X to be the column Hilbert space $H\sb{c},$ an equivalence is established between the matrix regularity of a space and that of its operator dual.
Beginning with the fact that all $C\sp*$-algebras, and in fact all operator systems, are matrix regular, it is shown that all operator spaces of the form $X\sp* \otimes\sp{h} V \otimes\sp{h} X$ and CB(V,B(H)) are matrix regular whenever V is. Reverse implications are also shown in some cases. The replacement of B(H) by an injective von Neumann algebra R is explored, leading to more general results and some extra results regarding $R\sp\prime$-module projective tensor products. Complex interpolation is used to define operator space structures on the Schatten class spaces $S\sb{p}$ and the commutative $L\sb{p}$-spaces. These spaces are then shown to be matrix regular. Some generalized Schatten class spaces are also shown to be matrix regular.
Finally, as an application, an alternative proof is presented for the Christensen-Sinclair Multilinear Representation Theorem that depends on matrix regularity rather than on Wittstock's complicated concept of matricial sublinearity.
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