Independent functions and the geometry of Banach spaces

Johnson–Schechtman inequalities in the free probability theory

“…If the algebra M coincides with either L ∞ (0, 1) or L ∞ (0, ∞), then the notion of singular value function coincides with the classical notion of decreasing rearrangement (see e.g. [11,3]). …”

“…In this (commutative) setting, the best possible results have been achieved in [1] (see also [2,4]) on the basis of operator approach introduced in [2] (see a detailed exposition in [3]). This approach has permitted to prove that the estimates (2) hold if and only if the so-called Kruglov operator defined in [2] maps the space E into itself.…”

“…Our approach is based on the study of the "free Kruglov operator" (see Section 4 below) and is of interest in its own right. It is based on earlier probabilistic ideas of Kruglov [15] and their adaptation to (commutative) symmetric function spaces by Braverman [5] and by Astashkin and the first-named author [1][2][3][4].…”

Embeddings of operator ideals intoLp-spaces on finite von Neumann algebras

“…Note that the methods used in [11,12,19,14,15] depend heavily on the techniques related to the theory of random processes. In a recent paper [8], a different approach based on methods and ideas from the interpolation theory of operators and the usage of so-called Kruglov operator [4,5,6,7] is suggested.…”

Johnson–Schechtman and Khintchine inequalities in noncommutative probability theory

“…Here, n k=0 f k := n k=0 f k (· − k)χ (k,k+1) is a disjoint sum of independent mean zero random variables {f k } n k=0 , which is a Lebesgue measurable function on (0, ∞). In the commutative case, the best possible results were achieved in [2,5] by using the so-called Kruglov operator/property introduced in [1,10] (see detailed exposition of this theme in [3]).…”