Factorization of \(p\)-completely bounded multilinear maps (Q1919892)

Given Banach spaces \(X_1,\dots, X_N\), \(Y_1,\dots, Y_N\), \(X\), \(Y\) and subspaces \(S_i\subset B(X_i, Y_i)\) \((1\leq i\leq N)\), we study \(p\)-completely bounded multilinear maps \(A: S_N\times\cdots\times S_1\to B(X, Y)\). We obtain a factorization theorem for such \(A\) which is entirely similar to the Christensen-Sinclair representation theorem for completely bounded multilinear maps on operator spaces. Our main tool is a generalization of Ruan's representation theorem for operator spaces in the Banach space setting. As a consequence, we are able to compute the norms of adapted multilinear Schur product maps on \(B(\ell^n_p)\).