## Non-commutative Burkholder/Rosenthal inequalities (2000)

Venue: | Ann. Probab |

Citations: | 52 - 27 self |

### BibTeX

@TECHREPORT{Junge00non-commutativeburkholder/rosenthal,

author = {M. Junge and Q. Xu},

title = {Non-commutative Burkholder/Rosenthal inequalities},

institution = {Ann. Probab},

year = {2000}

}

### OpenURL

### Abstract

Abstract. We show norm estimates for the sum of independent random variables in noncommutative Lp spaces for 1 < p < ∞ following previous work by the authors. These estimates generalize Rosenthal’s inequalities in the commutative case. Among other applications, we derive a formula for p-norm of the eigenvalues for matrices with independent entries, and characterize those symmetric subspaces and unitary ideal spaces which can be realized as subspaces of noncommutative Lp for 2 < p < ∞. 0. Introduction and Notation Martingale inequalities have a long tradition in probability. The applications of the work of Burkholder and his collaborators [B73,?, BDG72, B71a, B71b, BGS71, BG70, B66] ranges from classical harmonic analysis to stochastical differential equations and the geometry of Banach spaces. When proving the estimates for the ‘little square function ’ Burkholder

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Citation Context ...N,E;ℓ c 2 ), �(xk)�Lp(N,E;ℓ r 2 )} . max{�(xk)�Lp(N,E;ℓ c 2 ), �(xk)�Lp(N,E;ℓ r 2 )} ≤ = 1−θ ∞ � n� � n� xk�p ≤ C1p k=1 k=1 �xk� p p � n� � 1 p k=1 . �xk� p p � 1 p θ + . From the reiteration theorem =-=[BL]-=-, we deduce that 2 Lp(N; ℓ n p) = [Lp(N; ℓ n ∞), Lp(N; ℓ n 2)]θ . .sThis implies NONCOMMUTATIVE BURKHOLDER/ROSENTHAL INEQUALITIES II: APPLICATIONS 23 � n� k=1 k=1 �xk� p p � 1 p ≤ �(xk)� 1−θ Lp(N;ℓ n ... |

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Citation Context ...Haagerup trace. We will frequently use Hölder’s inequality whenever 1 p = 1 r + 1 q �xy�p ≤ �x�r�y�q . We say that M ⊂ N is a φ-invariant subalgebra, if σφ t (M) ⊂ M. According to Takesaki’s Theorem (=-=[Tak72]-=-), there exists a unique conditional expectationsNONCOMMUTATIVE BURKHOLDER/ROSENTHAL INEQUALITIES II: APPLICATIONS 5 E : N → M such that φ|M ◦ E = φ. This implies in particular (1.1) σ φ t ◦ E = E ◦ σ... |

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Citation Context ...N = Nn. The space Lp(N) = (1 − EM)Lp(N) is 2-complemented in Lp(N) and therefore the lower estimate � n� k=1 �xk�p � 1 p ≤ 2� follows by interpolation (using the inclusion map x → xD 1 2 ∈ L2(N), see =-=[Kos]-=- for more details on these interpolation spaces). Moreover, for x = � n k=1 xk, we have � n� k=1 n� k=1 xk�p EM(x ∗ kxk)� p 2 = �EM(x ∗ x)� p 2 ≤ �x∗ x� p 2 = �x�2 p . Therefore the lower estimates fo... |

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Citation Context ...ences (xk)1≤k≤n, n ∈ N, by and �x�Lp(N;ℓn ∞ ) = inf xk = aykb �a�2p sup k �x�Lp(N;ℓn 1 ) = inf xk = � a j ∗ kjbkj � � kj a ∗ kjakj� 1 p �yk��b�2p p � � kj b ∗ kjbkj� 1 p p . For more information (see =-=[Jun1]-=-) and [JRX2] where the connection to decomposable maps is explained). We may then define Lp(N; ℓ n q ) = [Lp(N; ℓ n ∞), Lp(N; ℓ n 1)] 1 q We will write Lp(N; ℓ n q ) if we want to emphasize the underl... |

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Citation Context ...ical harmonic analysis to stochastical differential equations and the geometry of Banach spaces. When proving the estimates for the ‘little square function’ Burkholder was aware of Rosenthal’s result =-=[Ros]-=- on sums of independent random variables. Here we proceed differently and prove the noncommutative Rosenthal inequality along the same line as the noncommutative Burkholder inequality from [JX1]. This... |

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Citation Context ...f the integers. This problem (even for scalar coefficients) is open for 1 ≤ p < 2. The problem is also open for 2 < p < ∞ without assuming that N is hyperfinite. On the Banach space level we refer to =-=[RX]-=- and [Ran] for different versions of the Kadec-Pe̷lcziński alternative. We will now show that conversely the only symmetric subspaces of Lp(N) are the one found in (5.3). The next result is a our star... |

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Citation Context .... JUNGE AND Q. XU for all xk ∈ Lq(Ak) with mean 0. Our aim is to prove the estimate for p. Of course we may assume p > 2. Let xk ∈ Lp(Ak) and EM(xk) = 0. First we apply the Khintchine inequality (see =-=[PS]-=- for the right order of constants) and deduce from Lemma 1.1 that (1.4) � n� xk�p ≤ 2 E� k=1 n� k=1 √ εkxk�p ≤ 2c1 p max{� n� x ∗ kxk� 1 2 , � k=1 p 2 n� xkx ∗ k� 1 2 } . Let us consider the first one... |

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Citation Context ...emented subspaces of Sp. Remark 6.4. Let X be an rearrangement invariant space on (0, ∞) and N be a semifinite von Neumann algebra with a normal faithful trace. Then one can define LX(N, τ) (see e.g. =-=[DDdP]-=-). The same argument as above shows that if LX(N, τ) is isomorphic to a subspace Y ⊂ Lp(M), then LX(N, τ) is isomorphic to Lp(N, τ), L2(N, τ) or Lp(N, τ) ∩ L2(N, τ). In [Jun3] we show that indeed all ... |

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Citation Context ...DER/ROSENTHAL INEQUALITIES II: APPLICATIONS 25 Indeed, P is a the projection onto a submodule (see [Lan] for more details, see also [JS] for a general treatment without assuming N∗ separable based on =-=[Pas]-=-). Thus for 2 ≤ p < ∞ the map up extends to an isometric isomorphism and P extends to a contractive projection from Lp(M, ℓc 2) to the image up. In [Jun1] we defined Lp(N, EM) as the closure of ND 1 p... |

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Citation Context .... fk(z) = agkb = a( � v ∗ kjwkj)b where (gk) ∈ L∞(N; ℓ n 1). Applying the continuous selection theorem again we find continuous maps a, b and y on 1 + iR such that kj fk(z) = a(z)yk(z)b(z) . We apply =-=[PX2]-=- and obtain an analytic invertible function α, β : Ω → L2p(N) such that α(z)α(z) ∗ = a(z)a(z) ∗ + δ1 and β(z) ∗ β(z) = b(z) ∗ b(z) + δ1 for all z ∈ ∂Ω. Then we may define yk(z) = α(z) −1 fk(z)β(z) −1 ... |

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Citation Context ..., P (xk)〉 = 〈P (xk), P (xk)〉 ≤ 〈(xk), (xk)〉 .sNONCOMMUTATIVE BURKHOLDER/ROSENTHAL INEQUALITIES II: APPLICATIONS 25 Indeed, P is a the projection onto a submodule (see [Lan] for more details, see also =-=[JS]-=- for a general treatment without assuming N∗ separable based on [Pas]). Thus for 2 ≤ p < ∞ the map up extends to an isometric isomorphism and P extends to a contractive projection from Lp(M, ℓc 2) to ... |

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Citation Context ... conjecture that there is is continuous passage for p → 1 improving the constant above. (2) The same proof works for q-Araki Wood factors defined by Shlyahktenko for q = 0 and for q ∈ (−1, 1) by Hiai =-=[Hia]-=-. For q = 1 we refer to [Jun4] for the appropriate gaussian substitute. (3) Using this Khintchine type inequality 4.5, we can deduce, as in [Jun4], that every quotient of Rp ⊕ Cp completely embeds int... |

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Citation Context ...∗ jkwjkb�1 � 1 2 w ∗ jkwjkbb ∗ � 1 2 ) Before we prove a stability result with respect to conditional expectations, we have to recall some facts about the Stinespring dilation theorem (see also [Jun1]=-=[JRX1]-=-). Following [Ru] there exists a normal representation π : M → N ¯⊗B(ℓ2) such that E(x) = (1 ⊗ e11)π(x)(1 ⊗ e11) .s16 M. JUNGE AND Q. XU (In the case of non-separable predual, we have to use a larger ... |

9 |
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Citation Context ...ed they satisfy appropriate independence conditions, for example if they are freely � 1 p .sNONCOMMUTATIVE BURKHOLDER/ROSENTHAL INEQUALITIES II: APPLICATIONS 3 independent (in the sense of Voiculescu =-=[VDN]-=-). We also extend our result to the nontracial setting and even non-faithful setting. The dual version of Rosenthal’s inequality (in the non-tracial setting) provides Khintchine type inequalities for ... |

6 |
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Citation Context ...ℓn ∞) ∗ , Lp ′(N; ℓn 1) ∗ ] 1 p ′ = [Lp ′(N; ℓn ∞) ∗ , Lp ′(N; ℓn 1) ∗ ] 1 p ′ which follows immediately from the fact that topological all these space coincide with (Lp ′(N)∗ ) n and Bergh’s theorem =-=[Be]-=-. For p = 1 the inclusion ℓ n 1(L1) ⊂ L1(ℓ n 1) follows from the fact that the space L1(ℓ n 1) is normed (see [Pis1] for a similar argument). Remark 3.5. Motivated from Pisier’s theory of vector value... |

6 |
Operator spaces, volume 23 of London Mathematical Society Monographs. New Series
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Citation Context ...e) probability and the theory of von Neumann algebras (see e.g. [KRI, KRII], [SZ, Str] and [TakI, TakII, TakIII]). For more information and definitions on operator space theory we refer to [Pis2] and =-=[ER]-=-. We refer to [Pis1, JNRX] for the natural operator space structure on Lp spaces. 1. Rosenthal’s inequality for sums of independent random variables In this section, we shall prove a noncommutative ve... |

6 |
Haagerup’s reduction on noncommutative Lp-spaces and aplications
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Citation Context ... = N ⋊ φ σ G where G = t � n 2−nZ is the group of duadic rationals. There is a conditional expectation F : N → N given by the identity element. Then ˆ φ = φ◦F is a normal faithful state. We refer for =-=[JX3]-=- for the fact that there is an increasing sequence Nm of ˆ φ invariant finite von Neumann algebras such that the corresponding conditional expectations Em : N → Nm converge to the identity in the stro... |

6 |
Type decomposition and the rectangular AFD property for W*-TRO’s
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Citation Context ...∗ jkwjkbb ∗ � 1 2 ) Before we prove a stability result with respect to conditional expectations, we have to recall some facts about the Stinespring dilation theorem (see also [Jun1][JRX1]). Following =-=[Ru]-=- there exists a normal representation π : M → N ¯⊗B(ℓ2) such that E(x) = (1 ⊗ e11)π(x)(1 ⊗ e11) .s16 M. JUNGE AND Q. XU (In the case of non-separable predual, we have to use a larger index set). There... |

6 | Operator-space Grothendieck inequalities for noncommutative Lp-spaces - Xu |

4 |
COLp spaces - the local structure of non-commutative Lp spaces
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(Show Context)
Citation Context ...n [JMST]. It will be convenient to state these result in the Banach space and operator space setting. We refer to [Pis2] for a general background on operator spaces and completely bounded maps and to =-=[JNRX]-=- for the operator space structure of noncommutative Lp spaces. In this paper we focus on subspaces X, Y of Lp(N). In this situation the cb-norm of a linear map T : X → Y is given by �T �cb = �idLp(B(ℓ... |