An elementary approach to the category of locally convex operator spaces (Q2771247)

The paper proposes an approach to the concept of a locally convex operator space that is claimed to be more accessible than the existing introductions. In fact, the proposed approach is essentially a repackaging of Ruan's well-known characterization of abstract operator spaces as matricially normed spaces satisfying two additional axioms [\textit{Z.-J. Ruan}, J. Funct. Anal. 76, 217-230 (1988; Zbl 0646.46055)]. The difference is that instead of a family of matrix spaces \(M_n(X)\), where \(X\) is a \(\mathbb K\)-vector space, one considers a single space \(X_J^{op}\) of infinite matrices indexed by the set \(J\times J\) with entries from \(X\), at most finitely many of which don't vanish. The space \(X_J^{op}\), viewed as a bimodule over \({\mathbb K}_J^{op}\), is then equipped with a norm (or a family of seminorms) satisfying Ruan's two axioms. Seminorms are described as Minkowski's gauge functionals of absolutely matrix convex sets, an idea originating in [\textit{E. G. Effros} and \textit{S. Winkler}, J. Funct. Anal. 144, 117-152 (1997; Zbl 0897.46046)]. Finally, a locally convex operator space [\textit{E. G. Effros} and \textit{C. Webster}, NATO ASI Ser., Ser. C, Math. Phys. Sci. 495, 163-207 (1997; Zbl 0892.46065)] is defined as a vector space \(X\) together with a topology making \({\mathbb K}_J^{op}\) into a topological vector space and possessing a basis consisting of absolutely matrix convex sets. Morphisms between locally convex operator spaces are defined in a usual way, through amplifications of linear operators. The concepts of operator bornology and operator weak topology are then introduced and discussed. NEWLINENEWLINENEWLINEEven if it is not quite clear in what sense the new approach is more elementary than the by now traditional definition of an abstract operator space (as Ruan's theorem, which is left out of the present note, will have to be proved sooner or later!), it offers a fresh and interesting viewpoint.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00047].