International Congress of Mathematicians 2018 , 2018-08-01

Title : ( Haagerup--Pisier--Ringrose inequality )

Authors: Mohammad Sal Moslehian ,

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‎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 ‎\begin{align}\label{HPR}‎ ‎\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\|‎ ‎\end{align}‎ ‎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 ‎\begin{align*}‎ ‎\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\|‎. ‎\end{align*}‎


, Haagerup--Pisier--Ringrose inequality, unitarily invariant norm, C*-algebra
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author = {Sal Moslehian, Mohammad},
title = {Haagerup--Pisier--Ringrose inequality},
booktitle = {International Congress of Mathematicians 2018},
year = {2018},
location = {ریودوژانیرو},
keywords = {Haagerup--Pisier--Ringrose inequality; unitarily invariant norm; C*-algebra},


%0 Conference Proceedings
%T Haagerup--Pisier--Ringrose inequality
%A Sal Moslehian, Mohammad
%J International Congress of Mathematicians 2018
%D 2018