Haagerup--Pisier--Ringrose inequality

Abstract

‎The Haagerup--Pisier--Ringrose inequality states that if $\\Phi$ is a bounded linear map from a $C^*$-algebra $\\mathscr{A}$ into a $C^*$-algebra $\\mathscr{B}$‎, ‎then‎<br><br>‎\\begin{align}\\label{HPR}‎<br><br>‎\\left\\|\\sum_{j=1}^n\\left\\{\\Phi(A_j)^* \\Phi(A_j)+\\Phi(A_j) \\Phi(A_j)^*\\right\\} \\right\\|\\leq K\\|\\Phi\\|^2 \\left\\|\\sum_{j=1}^n\\left(\\sqsq{A_j}\\right)\\right\\|‎<br><br>‎\\end{align}‎<br><br>‎holds for $K=4$ and for any finite family $\\{A_1‎, ‎\\cdots‎, ‎A_n\\}$ of elements of $\\mathscr{A}$‎. ‎In this talk‎, ‎we present several versions of the Haagerup--Pisier--Ringrose inequality involving unitarily invariant norms and unital completely positive maps‎. ‎Among other things‎, ‎we show that if $\\mathcal{J}$ is the ideal of $\\mathbb{B}(\\mathscr{H})$ associated to a unitarily invariant norm $

|\\cdot

|$‎, ‎$\\Phi‎: ‎\\mathscr{A} \\to \\mathscr{B}$ is a bounded linear map between $C^*$-algebras‎, ‎$A_1‎, ‎\\cdots‎, ‎A_n\\in \\mathscr{A}$ are positive such that $\\Phi(A_j)$'s commute for $1\\leq j\\leq n$ and $X_1‎, ‎\\cdots‎, ‎X_n \\in \\mathcal{J}$‎. ‎Then‎<br><br>‎\\begin{align*}‎<br><br>‎\\left|\\left|\\left| \\sum_{j=1}^n\\left\\{ \\Phi(A_j)X_j\\Phi(A_j)^*+\\Phi(A_j)^*X_j\\Phi(A_j)\\right\\}\\right|\\right|\\right|\\leq 4

|X_1 \\oplus \\cdots\\oplus X_n

|\\‎, ‎\\|\\Phi\\|^2\\left\\|\\sum_{j=1}^n A_j^2\\right\\|‎.<br><br>‎\\end{align*}‎