Isometric embeddability of [formula omitted] into [formula omitted

Abstract

In this paper, we study existence of isometric embedding of Sqm into Spn, where 1≤p≠q≤∞ and n≥m≥2. We show that for all n≥m≥2 if there exists a linear isometry from Sqm into Spn, where (q,p)∈(1,∞]×(1,∞)∪(1,∞)∖{3}×{1,∞} and p≠q, then we must have q=2. This mostly generalizes a classical result of Lyubich and Vaserstein. We also show that whenever Sq embeds isometrically into Sp for (q,p)∈(1,∞)×[2,∞)∪[4,∞)×{1}∪{∞}×(1,∞)∪[2,∞)×{∞} with p≠q, we must have q=2. Thus, our work complements work of Junge, Parcet, Xu and others on isometric and almost isometric embedding theory on non-commutative Lp-spaces. Our methods rely on several new ingredients related to perturbation theory of linear operators, namely Kato-Rellich theorem, theory of multiple operator integrals and Birkhoff-James orthogonality, followed by thorough and careful case by case analysis. The question whether for m≥2 and 1<q<2, Sqm embeds isometrically into S∞n, was left open in Bull. London Math. Soc. 52 (2020) 437-447.