Acta mathematica scientia,Series B ›› 2021, Vol. 41 ›› Issue (6): 2173-2182.doi: 10.1007/s10473-021-0622-2

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Marius JUNGE1, Quanhua XU2   

  1. 1. Department of Mathematics, University of Illinois, Urbana, IL 61801, USA;
    2. Institute for Advanced Study in Mathematics, Harbin Institute of Technology, Harbin 150001, China;Laboratoire de Mathématiques, Université de Bourgogne Franche-Comté, 25030 Besançon Cedex, France
  • Received:2021-04-14 Revised:2021-10-15 Online:2021-12-25 Published:2021-12-27
  • Supported by:
    Xu was partially supported by the French ANR project (ANR-19-CE40-0002) and the Natural Science Foundation of China (12031004).

Abstract: Let $\mathcal{M}$ be a semifinite von Neumann algebra. We equip the associated noncommutative $L_p$-spaces with their natural operator space structure introduced by Pisier via complex interpolation. On the other hand, for $1 < p < \infty$ let $$L_{p,p}(\mathcal{M})=\big(L_{\infty}(\mathcal{M}),\,L_{1}(\mathcal{M})\big)_{\frac1p,\,p}$$ 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 $L_{p,p}(\mathcal{M})=L_{p}(\mathcal{M})$ completely isomorphically if and only if $\mathcal{M}$ 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 $1 < p < \infty$ and $1\le q\le\infty$ with $p\neq q$ $$\big(L_{\infty}(\mathcal{M};\ell_q),\,L_{1}(\mathcal{M};\ell_q)\big)_{\frac1p,\,p}=L_p(\mathcal{M}; \ell_q)$$ with equivalent norms, i.e., at the Banach space level if and only if $\mathcal{M}$ is isomorphic, as a Banach space, to a commutative von Neumann algebra. \\ Our third result concerns the following inequality: $$ \Big\|\big(\sum_ix_i^q\big)^{\frac1q}\Big\|_{L_p(\mathcal{M})}\le \Big\|\big(\sum_ix_i^r\big)^{\frac1r}\Big\|_{L_p(\mathcal{M})} $$ for any finite sequence $(x_i)\subset L_p^+(\mathcal{M})$, where $0 < r < q < \infty$ and $0 < p\le\infty$. If $\mathcal{M}$ is not isomorphic, as a Banach space, to a commutative von Meumann algebra, then this inequality holds if and only if $p\ge r$.

Key words: operator spaces, Lp-spaces, real interpolation, column Hilbertian spaces

CLC Number: 

  • 46E30