TY - JOUR

T1 - Notes on Real Interpolation of Operator Lp-Spaces

AU - Junge, Marius

AU - Xu, Quanhua

N1 - Funding Information:
Xu was partially supported by the French ANR project (ANR-19-CE40-0002) and the Natural Science Foundation of China (12031004).
Publisher Copyright:
© 2021, Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences.

PY - 2021/11

Y1 - 2021/11

N2 - Let ℳ be a semifinite von Neumann algebra. We equip the associated non-commutative Lp-spaces with their natural operator space structure introduced by Pisier via complex interpolation. On the other hand, for 1 < p < ∞ let Lp,p(ℳ)=(L∞(ℳ),L1(ℳ))1p,p be equipped with the operator space structure via real interpolation as defined by the second named author (J. Funct. Anal. 139 (1996), 500-539). We show that Lp,p(ℳ) = Lp(ℳ) completely isomorphically if and only if ℳ is finite dimensional. This solves in the negative the three problems left open in the quoted work of the second author. We also show that for 1 < p < ∞ and 1 ≤ q ≤ ∞ with p ≠ q(L∞(ℳ;ℓq),L1(ℳ;ℓq))1p,p=Lp(ℳ;ℓq) with equivalent norms, i.e., at the Banach space level if and only if ℳ is isomorphic, as a Banach space, to a commutative von Neumann algebra. Our third result concerns the following inequality: ‖(∑ixiq)1q‖Lp(ℳ)≤‖(∑ixir)1r‖Lp(ℳ) for any finite sequence (xi)⊂Lp+(ℳ), where 0 < r < q < ∞ and 0 < p ≤ ∞. If ℳ is not isomorphic, as a Banach space, to a commutative von Meumann algebra, then this inequality holds if and only if p ≥ r.

AB - Let ℳ be a semifinite von Neumann algebra. We equip the associated non-commutative Lp-spaces with their natural operator space structure introduced by Pisier via complex interpolation. On the other hand, for 1 < p < ∞ let Lp,p(ℳ)=(L∞(ℳ),L1(ℳ))1p,p be equipped with the operator space structure via real interpolation as defined by the second named author (J. Funct. Anal. 139 (1996), 500-539). We show that Lp,p(ℳ) = Lp(ℳ) completely isomorphically if and only if ℳ is finite dimensional. This solves in the negative the three problems left open in the quoted work of the second author. We also show that for 1 < p < ∞ and 1 ≤ q ≤ ∞ with p ≠ q(L∞(ℳ;ℓq),L1(ℳ;ℓq))1p,p=Lp(ℳ;ℓq) with equivalent norms, i.e., at the Banach space level if and only if ℳ is isomorphic, as a Banach space, to a commutative von Neumann algebra. Our third result concerns the following inequality: ‖(∑ixiq)1q‖Lp(ℳ)≤‖(∑ixir)1r‖Lp(ℳ) for any finite sequence (xi)⊂Lp+(ℳ), where 0 < r < q < ∞ and 0 < p ≤ ∞. If ℳ is not isomorphic, as a Banach space, to a commutative von Meumann algebra, then this inequality holds if and only if p ≥ r.

KW - 46E30

KW - 47C15

KW - column Hilbertian spaces

KW - L-spaces

KW - operator spaces

KW - real interpolation

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U2 - 10.1007/s10473-021-0622-2

DO - 10.1007/s10473-021-0622-2

M3 - Article

AN - SCOPUS:85118742850

VL - 41

SP - 2173

EP - 2182

JO - Acta Mathematica Scientia

JF - Acta Mathematica Scientia

SN - 0252-9602

IS - 6

ER -