Non-commutative Khintchine type inequalities associated with free groups
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Let F n denote the free group with n generators g 1 , g 2 , . . . , g n . Let stand for the left regular representation of F n and let 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 () generated by the family of operators (gi 1 gi 2 gi d ) 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 () generated by the image under 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 ( ij a* ij a ij ) 1/2 and ( 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 , we have K p (K 2 , K ) with 1/p = (1 - )/2. Indiana University Mathematics Journal .