Factorization of p-completely bounded multilinear maps

Christian Le Merdy

Vol. 172 (1996), No. 1, 187–213

DOI: 10.2140/pjm.1996.172.187

Abstract

Given Banach spaces X₁,…,X_N, Y ₁,…,Y _N, X, Y and subspaces S_i ⊂ B(X_i,Y _i) (1 ≤ i ≤ N), we study p-completely bounded multilinear maps A : S_N ×⋯ × S₁ → 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 generalisation 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(ℓ_pⁿ).

Mathematical Subject Classification 2000

Primary: 46L05

Secondary: 46M05, 47B99

Milestones

Received: 14 July 1993

Revised: 22 September 1993

Published: 1 January 1996