Strong Haagerup inequalities for free R-diagonal elements

Multivariate Trace Inequalities, p-Fidelity, and Universal Recovery Beyond Tracial Settings

Trace inequalities are general techniques with many applications in quantum information theory, often replacing classical functional calculus in noncommutative settings. The physics of quantum field theory and holography, however, motivate entropy inequalities in type III von Neumann algebras that lack a semifinite trace. The Haagerup and Kosaki $L_p$ spaces enable re-expressing trace inequalities in non-tracial von Neumann algebras. In particular, we show this for the generalized Araki-Lieb-Thirring and Golden-Thompson inequalities from (Sutter, Berta \& Tomamichel 2017). Then, using the Haag…

Brown measure of $R$-diagonal operators, revisited

We use subordination functions perspective to reformulate Haagerup--Schultz's approach for the Brown measure of $R$-diagonal operators. This allows us to simplify the original argument and find a connection with the other approach due to Belinschi-Śniady-Speicher. The Brown measure formula can be rewritten in terms of subordination functions.

Multivariable Bohr inequalities

Operator-valued multivariable Bohr type inequalities are obtained for: a class of noncommutative holomorphic functions, generalizing the analytic functions on the open unit disc; the noncommutative disc algebra and the noncommutative analytic Toeplitz algebra; a class of noncommutative selfadjoint harmonic functions, generalizing the real-valued harmonic functions on the open unit disc; Cuntz-Toeplitz algebras ; the reduced (resp.full) group C*-algebra of the free group with n generators; a class of analytic functions on the open unit ball of C^n. The classical Bohr inequality is shown to be a…

关于 Grothendieck 不等式的若干观点

Some points of view on Grothendieck's inequalities

本文基于 Haagerup 的非交换小 Grothendieck 不等式证明，利用完全有界映射理论揭示交换 Grothendieck 不等式与小不等式的几何联系，并刻画复情形的小常数。

Strong Haagerup inequality with operator coefficients

We prove a Strong Haagerup inequality with operator coefficients. If for an integer d, H_d denotes the subspace of the von Neumann algebra of a free group F_I spanned by the words of length d in the generators (but not their inverses), then we provide in this paper an explicit upper bound on the norm on M_n(H_d), which improves and generalizes previous results by Kemp-Speicher (in the scalar case) and Buchholz and Parcet-Pisier (in the non-holomorphic setting). Namely the norm of an element of the form $\sum_{i=(i_1,..., i_d)} a_i \otimes λ(g_{i_1} ... g_{i_d})$ is less than $4^5 \sqrt e (\|M_…

The One Dimensional Free Poincaré Inequality

In this paper we discuss the natural candidate for the one dimensional free Poincaré inequality. Two main strong points sustain this candidacy. One is the random matrix heuristic and the other the relations with the other free functional inequalities, namely, the free transportation and Log-Sobolev inequalities. As in the classical case the Poincaré is implied by the others. This investigation is driven by a nice lemma of Haagerup which relates logarithmic potentials and Chebyshev polynomials. The Poincaré inequality revolves around the counting number operator for the Chebyshev polynomials of…

Operator space Grothendieck inequalities for noncommutative $L_p$-spaces

We prove the operator space Grothendieck inequality for bilinear forms on subspaces of noncommutative $L_p$-spaces with $2<p<\infty$. One of our results states that given a map $u: E\to F^*$, where $E, F\subset L_p(M)$ ($2<p<\infty$, $M$ being a von Neumann algebra), $u$ is completely bounded iff $u$ factors through a direct sum of a $p$-column space and a $p$-row space. We also obtain several operator space versions of the classical little Grothendieck inequality for maps defined on a subspace of a noncommutative $L_p$-space ($2<p<\infty$) with values in a $q$-column space for every $q\in [p'…

Diagonals of self-adjoint operators

The eigenvalues of a self-adjoint nxn matrix A can be put into a decreasing sequence $λ=(λ_1,...,λ_n)$, with repetitions according to multiplicity, and the diagonal of A is a point of $R^n$ that bears some relation to $λ$. The Schur-Horn theorem characterizes that relation in terms of a system of linear inequalities. We give a new proof of the latter result for positive trace-class operators on infinite dimensional Hilbert spaces, generalizing results of one of us on the diagonals of projections. We also establish an appropriate counterpart of the Schur inequalities that relate spectral proper…

非交换对称空间中的非对称Burkholder不等式

Asymmetric Burkholder inequalities in noncommutative symmetric spaces

本文在对称算子空间中建立了非对称Burkholder不等式，通过复插值和对偶方法证明，并推广了Johnson-Schechtman不等式。

算子矩阵数值半径不等式的改进上界及其应用

Improved Upper Bounds for Numerical Radius Inequalities of Operator Matrices and Their Applications

本研究运用多种不等式及Buzano不等式扩展形式，推导希尔伯特空间上算子矩阵数值半径的改进上界，完善经典结果，应用于量子力学等多领域的数学物理分析。

非交换 $L_p$-空间中的锐切向不等式

Sharp tangent inequalities in noncommutative $L_p$-spaces

本文在非交换 $L_p$-空间（Haagerup 空间）中建立了锐切向不等式，推广了 Xu 的经典结果，并给出 Ricard--Xu 与 Ball--Carlen--Lieb 凸性不等式的切向形式，常数 $p-1$ 最优，还补充了 $0<p<1$ 时的正性情形。

On spectra and Brown's spectral measures of elements in free products of matrix algebras

We compute spectra and Brown measures of some non self-adjoint operators in $(M_2(\cc), {1/2}Tr)*(M_2(\cc), {1/2}Tr)$, the reduced free product von Neumann algebra of $M_2(\cc)$ with $M_2(\cc)$. Examples include $AB$ and $A+B$, where A and B are matrices in $(M_2(\cc), {1/2}Tr)*1$ and $1*(M_2(\cc), {1/2}Tr)$, respectively. We prove that AB is an R-diagonal operator (in the sense of Nica and Speicher \cite{N-S1}) if and only if Tr(A)=Tr(B)=0. We show that if X=AB or X=A+B and A,B are not scalar matrices, then the Brown measure of X is not concentrated on a single point. By a theorem of Haagerup…