Matrix Regular Operator Spaces

Abstract

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 spaceEis regular if: (1) −x⩽y⩽ximplies that ‖y‖⩽‖x‖; and (2) ‖y‖<1 implies the existence ofx∈Esuch that ‖x‖<1 and −x⩽y⩽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. 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 whereVis a matrix ordered operator space, a natural cone is defined on the operator spaceX*⊗hV⊗hX, with ⊗hindicating the Haagerup tensor product, so as to make it a matrix ordered operator space. Exploiting the advantages gained by takingXto be the column Hilbert spaceHc, an equivalence is established between the matrix regularity of a space and that of its operator dual. This concept of matrix regularity also provides for more accessible proofs of the Christensen–Sinclair Multilinear Representation and Multilinear Decomposition theorems.