Non-commutative Khintchine type inequalities associated with free groups
Non-commutative Khintchine type inequalities associated with free groups
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与自由群相关的非交换辛钦型不等式
DOI：
10.1512/iumj.2005.54.2612
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发表时间：
2003
影响因子：
1.1
通讯作者：
G. Pisier
中科院分区：
文献类型：
--
作者：
Javier Parcet;G. Pisier
设Fn表示具有n个生成元g1，g2，...的自由群，g n .设A为Fn的左正则表示，T为A的标准迹.给定任意正整数d，研究Lp（τ）中由算子族λ（gi，g2. g id），其中1 < i k ≤ n。此外，我们对这个算子空间的描述保持了一个不依赖于n或p的常数，因此我们的结果对无穷多个生成元仍然有效。我们还考虑了Lp（τ）中由长度为d的缩减字集A下的像生成的子空间。我们的结果推广了布赫霍尔茨关于空间W∞（n，d）的一个结果到任意指数1 < p < ∞.主要的应用是一定的插值定理，有效的任何程度d（延长结果的第二作者，限于d = 1）。在d = 2的最简单情况下，我们的定理可以表述如下：考虑由所有在Schatten类Sp中有元素的分块矩阵a =（a ij）构成的空间Kp，使得a在Sp中相对于l 2 <$l 2，并且，使得（ij a* ij a ij）1/2和（ij a ij a* ij）1/2都属于Sp。我们用三个相应范数的最大值来赋予K p。然后，对于2 ≤ p ≤ oo，我们有K p（K 2，K∞）θ，其中1/p =（1 - θ）/2。
Let F n denote the free group with n generators g 1 , g 2 ,..., g n . Let A stand for the left regular representation of F n and let T be the standard trace associated to A. Given any positive integer d, we study the operator space structure of the subspace W p (n, d) of L p (τ) generated by the family of operators λ(g i ,g 2 ... g id ) with 1 < i k ≤ n. Moreover, our description of this operator space holds up to a constant which does not depend on n or p, so that our result remains valid for infinitely many generators. We also consider the subspace of L p (τ) generated by the image under A of the set of reduced words of length d. Our result extends to any exponent 1 < p < ∞ a previous result of Buchholz for the space W∞(n,d). The main application is a certain interpolation theorem, valid for any degree d (extending a result of the second author, restricted to d = 1). In the simplest case d = 2, our theorem can be stated as follows: consider the space K p formed of all block matrices a = (a ij ) with entries in the Schatten class S p , such that a is in S p relative to l 2 ⊗ l 2 and, moreover, such that (Σ ij a* ij a ij ) 1/2 and (Σ ij a ij a* ij ) 1/2 both belong to S p . We equip K p with the maximum of the three corresponding norms. Then, for 2 ≤ p ≤ oo, we have K p ≃ (K 2 , K∞)θ with 1/p = (1 - θ)/2.