The Chevet-Saphar tensor norms for operator spaces (Q2822579)

For \(1\leq p<\infty\), the author uses \(S_p\) to denote the Schatten \(p\)-class. Given operator spaces \(E\) and \(F\), a linear mapping \(u: E\to F\) is said to be completely \(p\)-summing if the mapping \(I_{S_p}\otimes u: S_P\otimes_ {\min}E\to S_p[F]\) is bounded. The author defines the right (resp., left) Chevet-Saphar operator tensor norm of order \(p\), \(d_P^o\) (resp., \(g_p^o\)), as the norm on \(E\otimes F\) so that the tensor contraction NEWLINE\[NEWLINE \kappa: (E\otimes_{\min}S_{p'})\otimes_{\text{proj}} S_p[F]\to E\otimes_{d_p^o}F NEWLINE\]NEWLINE NEWLINE\[NEWLINE (\text{resp., } \lambda: S_p[E]\otimes_{\text{proj}}(S_{p'}\otimes_{\min} F)\to E\otimes_{g_p^o}F) NEWLINE\]NEWLINE is a complete \(1\)-quotient. He shows that the dual of \(E\otimes_{d_p^o}F\) is completely isometrically isomorphic to the space of completely \(p\)-summing mappings from \(E\) into \(F'\). Both \(d_p^o\) and \(g_p^o\) are shown to be uniform tensor norms with \(d_p^o\) completely right projective and \(g_p^o\) completely left projective. Operator space variants of the Chevet-Persson-Saphar inequalities are proved, where \(\Delta_p\) is replaced with \(\Delta_p^o\), the operator space structure on \(S_p\otimes E\) inherited from \(S_P[E]\). Finally, the author characterises those operator spaces which are quotients of an ultrapower of \(S_p\) and shows that \(d_p^o\) or \(g_p^o\) can be used to give an alternative characterisation of completely \(p\)-summing operators.