NOTES ON REAL INTERPOLATION OF OPERATOR Lp-SPACES

Abstract

References 14

Metrics

doi:10.1007/s10473-021-0622-2

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} (M) = (L_{\infty} (M), L_{1} (M))_{\frac{1}{p}, p}

$L_{p,p}(\mathcal{M})=\big(L_{\infty}(\mathcal{M}),\,L_{1}(\mathcal{M})\big)_{\frac1p,\,p}$ 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} (M) = L_{p} (M)

$L_{p,p}(\mathcal{M})=L_{p}(\mathcal{M})$ completely isomorphically if and only if

M

$\mathcal{M}$ 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

$1 < p < \infty$ and

1 \leq q \leq \infty

$1\le q\le\infty$ with

p \neq q

$p\neq q$

(L_{\infty} (M; ℓ_{q}), L_{1} (M; ℓ_{q}))_{\frac{1}{p}, p} = L_{p} (M; ℓ_{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

M

$\mathcal{M}$ is isomorphic, as a Banach space, to a commutative von Neumann algebra. \\ Our third result concerns the following inequality:

‖ (\sum_{i} x_{i}^{q})^{\frac{1}{q}} ‖_{L_{p} (M)} \leq ‖ (\sum_{i} x_{i}^{r})^{\frac{1}{r}} ‖_{L_{p} (M)}

$\Big\|\big(\sum_ix_i^q\big)^{\frac1q}\Big\|_{L_p(\mathcal{M})}\le \Big\|\big(\sum_ix_i^r\big)^{\frac1r}\Big\|_{L_p(\mathcal{M})}$ for any finite sequence

(x_{i}) \subset L_{p}^{+} (M)

$(x_i)\subset L_p^+(\mathcal{M})$ , where

0 < r < q < \infty

$0 < r < q < \infty$ and

0 < p \leq \infty

$0 < p\le\infty$ . If

M

$\mathcal{M}$ is not isomorphic, as a Banach space, to a commutative von Meumann algebra, then this inequality holds if and only if

p \geq r

$p\ge r$ .

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

CLC Number:

46E30

Marius JUNGE, Quanhua XU. NOTES ON REAL INTERPOLATION OF OPERATOR L_p-SPACES[J].Acta mathematica scientia,Series B, 2021, 41(6): 2173-2182.

Trendmd

[1]	Bergh J, Löfström J. Interpolation Spaces. Berlin:Springer-Verlag, 1976
[2]	Effros Ed, Ruan Z-J. Operator Spaces. vol 23 of London Mathematical Society Monographs, New Series. New York:The Clarendon Press, Oxford University Press, 2000
[3]	Junge M. Doob's inequality for non-commutative martingales. J Reine Angew Math, 2002, 549:149-190
[4]	Junge M, Xu Q. Noncommutative maximal ergodic inequalities. J Amer Math Soc, 2007, 20:385-439
[5]	Junge M, Xu Q. Noncommutative Burkholder/Rosenthal inequalities II:Applications. Israel J Math, 2008, 167:227-282
[6]	Lust-Piquard F, Pisier G. Noncommutative Khintchine and Paley inequalities. Ark Mat, 1991, 29:241-260
[7]	Nhany M J L. La stabilité des espaces Lp non-commutatifs. Math Scand, 1997, 81:212-218
[8]	Pisier G. The operator Hilbert space OH, complex interpolation and tensor norms. Mem Amer Math Soc, 1996, 122(585)
[9]	Pisier G. Non-commutative vector valued Lp-spaces and completely p-summing maps. Astérisque, 1998, 247
[10]	Pisier G. Introduction to Operator Space Theory. vol 294 of London Mathematical Society Lecture Note Series. Cambridge:Cambridge University Press, 2003
[11]	Pisier G, Xu Q. Non-commutative Lp-spaces//Handbook of the Geometry of Banach Spaces, Vol 2. Amsterdam:North-Holland, 2003:1459-1517
[12]	Takesaki M. Theory of Operator Algebras I. Springer-Verlag, 1979
[13]	Xu Q. Interpolation of operator spaces. J Funct Anal, 1996, 139:500-539
[14]	Xu Q. Embedding of Cq and Rq into noncommutative Lp-spaces, 1≤ p < q ≤ 2. Math Ann, 2006, 335:109-131

Just accepted

Online first

Just accepted

Online first

Viewed

Full text

	From	local

	Times	5
	Rate	100%

Abstract

Just accepted	Online first	Issue

0	0	57

	From	Others

	Times	57
	Rate	100%

Cited

Web of Science	Crossref	ScienceDirect	Search for Citations in Google Scholar >>


This page requires you have already subscribed to WoS.

Shared

Discussed

[1]	Kwok-Pun HO. A GENERALIZATION OF BOYD'S INTERPOLATION THEOREM [J]. Acta mathematica scientia,Series B, 2021, 41(4): 1263-1274.
[2]	Peide LIU. DOOB'S MAXIMAL INEQUALITIES FOR MARTINGALES IN VARIABLE LEBESGUE SPACE [J]. Acta mathematica scientia,Series B, 2021, 41(1): 283-296.
[3]	Jae-Myoung KIM, Yun-Ho KIM, Jongrak LEE. RADIALLY SYMMETRIC SOLUTIONS FOR QUASILINEAR ELLIPTIC EQUATIONS INVOLVING NONHOMOGENEOUS OPERATORS IN AN ORLICZ-SOBOLEV SPACE SETTING [J]. Acta mathematica scientia,Series B, 2020, 40(6): 1679-1699.
[4]	L. E. LABUSCHAGNE, W. A. MAJEWSKI. DYNAMICS ON NONCOMMUTATIVE ORLICZ SPACES [J]. Acta mathematica scientia,Series B, 2020, 40(5): 1249-1270.
[5]	Jinxia LI, Baode LI, Jianxun HE. THE BOUNDEDNESS FOR COMMUTATORS OF ANISOTROPIC CALDERÓN-ZYGMUND OPERATORS [J]. Acta mathematica scientia,Series B, 2020, 40(1): 45-58.
[6]	Elhoussine AZROUL, Badr LAHMI, Ahmed YOUSSFI. STRONGLY NONLINEAR VARIATIONAL PARABOLIC EQUATIONS WITH p(x)-GROWTH [J]. Acta mathematica scientia,Series B, 2016, 36(5): 1383-1404.
[7]	Ha Duy HUNG, Luong Dang KY. NEW WEIGHTED MULTILINEAR OPERATORS AND COMMUTATORS OF HARDY-CESÁRO TYPE [J]. Acta mathematica scientia,Series B, 2015, 35(6): 1411-1425.
[8]	Ying WANG, Mingxin WANG. SOLUTIONS TO NONLINEAR ELLIPTIC EQUATIONS WITH A GRADIENT [J]. Acta mathematica scientia,Series B, 2015, 35(5): 1023-1036.
[9]	Abdugheni ABDUREXIT, Turdebek N. BEKJAN. NONCOMMUTATIVE ORLICZ-HARDY SPACES [J]. Acta mathematica scientia,Series B, 2014, 34(5): 1584-1592.
[10]	Serkan Demiriz, Celal Cakan. SOME TOPOLOGICAL AND GEOMETRICAL PROPERTIES OF THE SEQUENCE SPACE e^r(u, p) [J]. Acta mathematica scientia,Series B, 2012, 32(4): 1513-1528.
[11]	WANG Ying, LUO Dang, WANG Ming-Xin. POSITIVE SOLUTIONS TO THE p-LAPLACIAN WITH SINGULAR WEIGHTS [J]. Acta mathematica scientia,Series B, 2012, 32(3): 1002-1020.
[12]	HAN Ya-Zhou, Turdebek N. Bekjan. THE DUAL OF NONCOMMUTATIVE LORENTZ SPACES [J]. Acta mathematica scientia,Series B, 2011, 31(5): 2067-2080.
[13]	CHEN Wei, LIU Pei-De. SEVERAL WEAK-TYPE WEIGHTED INEQUALITIES IN ORLICZ MARTINGALE CLASSES [J]. Acta mathematica scientia,Series B, 2011, 31(3): 1041-1050.
[14]	B.Zlatanov . ON A CLASS OF KÖTHE SEQUENCE SPACES WITH NORMAL STRUCTURE [J]. Acta mathematica scientia,Series B, 2011, 31(2): 576-590.
[15]	Binod Chandra Tripathy, Bipul Sarma . SOME DOUBLE SEQUENCE SPACES OF FUZZY NUMBERS DEFINED BY ORLICZ FUNCTIONS [J]. Acta mathematica scientia,Series B, 2011, 31(1): 134-140.