Matrix regular operator spaces
A norm on an ordered real Banach space \(E\) is called regular (or Riesz norm) if \(-x\leq y\leq x\) implies \(\| y\|\leq\| x\|\), and \(\| y\|< 1\) implies the existence of \(x\in E\) with \(\| x\|< 1\) and \(-x\leq y\leq x\). This concept is generalized to matrix ordered complex operator spaces [as introduced by \textit{M.-D. Choi} and \textit{E. G. Effros}, J. Funct. Anal. 24, 156-209 (1977; Zbl 0341.46049)] in a way which respects the matricial structure of the operator space. Such a space is called matrix regular if, at each matrix level, the restriction of the norm to the selfadjoint elements is regular in the above sense. The author studies hereditary properties of this notion. In particular, he shows that, if \(X\), \(V\) are operator spaces and \(V\) is matrix regular, then so is \(X^*\otimes^h V\otimes^h X\), where \(\otimes^h\) denotes the Haagerup tensor product. As an application, the author presents new proofs of both the multilinear representation theorem and the multilinear decomposition theorem of \textit{E. Christensen} and \textit{A. M. Sinclair} [J. Funct. Anal. 72, 151-181 (1987; Zbl 0622.46040)].
matricial structure of the operator space
scientific article; zbMATH DE number 1121257
scientific article