On the complete bounds of $$L_p$$ L p -Schur multipliers

Abstract

We study the class Mp of Schur multipliers on the Schattenvon Neumann class Sp with 1 ≤ p ≤ ∞ as well as the class of completely bounded Schur multipliers M cb p. We first show that for 2 ≤ p < q ≤ ∞ there exists m ∈ M cb p with m ∈ Mq, so in particular the following inclusions that follow from interpolation are strict: Mq Mp and M cb q M cb p. In the remainder of the paper we collect computational evidence that for p = 1, 2, ∞ we have Mp = M cb p , moreover with equality of bounds and complete bounds. This would suggest that a conjecture raised by Pisier (Astérisque 247:vi+131, 1998) is false.

Multilinear transference of Fourier and Schur multipliers acting on noncommutative -spaces

“…However, as [JTT09, Lemma 3.3] shows, for Schur multipliers on , just boundedness on the Haagerup tensor product is sufficient to guarantee that is multiplicatively bounded. Note that even in the linear case, when , it is unknown whether a bounded Schur multiplier on is necessarily completely bounded unless has continuous symbol (we refer to [Pi98, Conjecture 8.1.12], [LaSa11, Theorem 1.19], and [CaWi19]).…”