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.
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