Notes on real interpolation of operator $L_p$-spaces

DOI10.1007/S10473-021-0622-2arXiv2107.08404MaRDI QIDQ6373067

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Publication date: 18 July 2021

Abstract: Let $mathcalM$ 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 ${\displaystyle }$L_{p,p}(mathcal{M})=�ig(L_{infty}(mathcal{M}),,L_{1}(mathcal{M})�ig)_{frac1p,,p}${\displaystyle }$ be equipped with the operator space structure via real interpolation as defined by the second named author ({em J. Funct. Anal}. 139 (1996), 500--539). We show that $L_{p,p}(mathcalM)=L_p(mathcalM)$ completely isomorphically if and only if $mathcalM$ 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 $1leqleinfty$ with $peqq$ ${\displaystyle }$�ig(L_{infty}(mathcal{M};ell_q),,L_{1}(mathcal{M};ell_q)�ig)_{frac1p,,p}=L_p(mathcal{M}; ell_q)${\displaystyle }$ with equivalent norms, i.e., at the Banach space level if and only if $mathcalM$ is isomorphic, as a Banach space, to a commutative von Neumann algebra. Our third result concerns the following inequality: ${\displaystyle }$ �ig|�ig(sum_ix_i^q�ig)^{frac1q}�ig|_{L_p(mathcal{M})}le�ig|�ig(sum_ix_i^r�ig)^{frac1r}�ig|_{L_p(mathcal{M})} ${\displaystyle }$ for any finite sequence $(x_i)subset{L}_p^{+}(mathcalM)$, where $0<r<q<infty$ and $0<pleinfty$. If $mathcalM$ is not isomorphic, as a Banach space, to a commutative von Meumann algebra, then this inequality holds if and only if $pger$.

Mathematics Subject Classification ID

Interpolation between normed linear spaces (46B70) Noncommutative function spaces (46L52)

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