On real operator spaces
This paper deals with real operator spaces. By definition, a real operator space is a (closed) subspace of \(B(H)\), the space of all bounded linear operators on a real Hilbert space. The author gives an abstract characterization of such spaces, similar to his famous theorem on the characterization of the usual (complex) operator spaces. He has transferred many notions and results in the complex case to the present real setting. These include injectivity, tensor products, and others. Concerning injectivity, the author shows that a real operator space is injective iff it is a finite direct sum of rectangular matrix spaces with entries in \(\mathbb R\), \(\mathbb C\) and \(\mathbb H\) (\(\mathbb H\) being the quaternion ring). He also obtains a representation theorem for completely bounded maps from a \(C^*\)-algebra into a \(B(H)\).
scientific article; zbMATH DE number 2002321