Non-commutative Khintchine type inequalities associated with free groups

Abstract

Let Fn denote the free group with n generators g 1 , g 2 , . . ., gn.Let λ stand for the left regular representation of Fn and let τ be the standard trace associated to λ.Given any positive integer d, we study the operator space structure of the subspace Wp(n, d) of Lp(τ ) generated by the family of operators λ(gMoreover, 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 Lp(τ ) generated by the image under λ 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 Kp formed of all block matrices a = (a ij ) with entries in the Schatten class Sp, such that a is in Sp relative to ℓ 2 ⊗ ℓ 2 and moreover such that ( ij a * ij a ij ) 1/2 and ( ij a ij a * ij ) 1/2 both belong to Sp.We equip Kp with the maximum of the three corresponding norms.Then, for 2 ≤ p ≤ ∞ we have Kp ≃ (K 2 , K∞) θ with 1/p = (1 -θ)/2.

Locations

• Indiana University Mathematics Journal - View
• arXiv (Cornell University) - View - PDF