\((n+1)\)-tensor norms of Lapresté's type (Q2902689)

The author studies an \((n + 1)\)-tensor norm \(\alpha_{\mathbf{ {r}}}\) (where \({\mathbf{ {r}}}=(r_0,r_1,r_2, \ldots, r_n,r_{n+1})\) is an \((n+2)\)-tuple) extending to \((n + 1)\)-fold tensor products the classical one of Lapresté in the case \(n = 1\). He characterises \({\mathbf{ {r}}}\)-dominated, \({\mathbf{ {r}}}\)-nuclear and \({\mathbf{ {r}}}\)-integral multilinear mappings, respectively. The results and proofs are similar to the ones for related \((n+1)\)-tensor norms (see, e.g. the author's article [Positivity\ 11, 95--117 (2007; Zbl 1128.46028)] on Saphar type \((n+1)\)-tensor norms), e.g., factorizations and ultrapowers are involved. As an application, the author gives a complete description of the reflexivity of the \(\alpha_{\mathbf{ {r}}}\)-tensor product (\(\bigotimes_{j=1}^{n+1} \ell^{u_{j}}, \alpha_{\mathbf{ {r}}}\)). This includes the classical case \(n=1\), for which this result seems to be new.