Hilbertian Matrix Cross Normed Spaces Arising from Normed Ideals

Abstract. Generalizing Pisier’s idea, we introduce a Hilbertian matrix cross normed space associated with a pair of symmetric normed ideals. When the two ideals coincide, we show that our construction gives an operator space if and only if the ideal is the Schatten class. In general, a pair of symmetric normed ideals that are not necessarily the Schatten class may give rise to an operator space. We study the space of completely bounded mappings between the matrix cross normed spaces obtained in this way and show that the multiplicator norm naturally appears as the completely bounded norm.