Free pluriharmonic majorants and noncommutative interpolation

In this paper, we initiate the study of sub-pluriharmonic curves in Cuntz-Toeplitz algebras and free pluriharmonic majorants on noncommutative balls. We are lead to a characterization of the noncommutative Hardy space $H^2_{\bf ball}$ in terms of free pluriharmonic majorants, and to a Schur type description of the unit ball of $H^2_{\bf ball}$. These results are used to solve a multivariable commutant lifting problem and provide a description of all solutions.Comment: 35 page

The forgotten monoid

We study properties of the forgotten monoid which appeared in work of Lascoux and Schutzenberger and recently resurfaced in the construction of dual equivalence graphs by Assaf. In particular, we provide an explicit characterization of the forgotten classes in terms of inversion numbers and show that there are n^2-3n+4 forgotten classes in the symmetric group S_n. Each forgotten class contains a canonical element that can be characterized by pattern avoidance. We also show that the sum of Gessel's quasi-symmetric functions over a forgotten class is a 0-1 sum of ribbon-Schur functions.Comment: 13 pages; in version 3 the proof of Proposition 3 is correcte

The little Grothendieck theorem and Khintchine inequalities for symmetric spaces of measurable operators

We prove the little Grothendieck theorem for any 2-convex noncommutative symmetric space. Let \M be a von Neumann algebra equipped with a normal faithful semifinite trace \t, and let $E$ be an r.i. space on (0, \8). Let E(\M) be the associated symmetric space of measurable operators. Then to any bounded linear map $T$ from E(\M) into a Hilbert space $\mathcal H$ corresponds a positive norm one functional f\in E_{(2)}(\M)^* such that \forall x\in E(\M)\quad \|T(x)\|^2\le K^2 \|T\|^2 f(x^*x+xx^*), where $E_{(2)}$ denotes the 2-concavification of $E$ and $K$ is a universal constant. As a consequence we obtain the noncommutative Khintchine inequalities for E(\M) when $E$ is either 2-concave or 2-convex and $q$-concave for some q<\8. We apply these results to the study of Schur multipliers from a 2-convex unitary ideal into a 2-concave one.Comment: 14 pages. To appear in J. Funct. Ana