Non-commutative BMO space
type: Item
instance of: scholarly article
title: Non-commutative BMO space
zbMATH Open document ID: 0967.47018
DOI: 10.1007/PL00000415
author: Nicolae Popa
published in: Archiv der Mathematik
publication date: 2001-08-27T00:00:00Z
review text: Let \(A\in B(\ell_2)\) having the representation as a matrix \(A=(a(i,j))_{i,j=1}^\infty\). By \(P_T\) denote the triangular projection \((P_TA)(i,j)=a(i,j)\), if \(1\leq i\leq j<\infty\) and \((P_TA)(i,j)=0\) otherwise, while \(P_n\) is the following projection: \((P_nA)(i,j)=a(i,j)\), if \(1\leq i,j\leq n\) and \((P_nA)(i,j)=0\) otherwise. Define \(\text{BMO}_G^B=\{A\in B(\ell_2); \sup_{n\geq 0} \|P_n|A_n-P_{n-1}A|^2\|_\infty<\infty\}\), where \(|A|^2=A^\ast A\) and let \(\text{BMO}_G (M)\) be the completetion of \(\text{BMO}_G^B\) with respect the norm above. Similarly, \(A\in \text{BMO}_R (M)\) if \(A^\ast\in \text{BMO}_G (M)\). Finally, \(\text{BMO} (M)=\text{BMO}_G (M)\cap \text{BMO}_R (M).\) The main result contains the following Theorem 1: \(P_T: B(\ell_2)\to \text{BMO} (M)\) is a bounded linear operator. The theorem is a non-commutative analogue of the well known fact that the Riesz projection maps \(L_\infty\) into BMO.
reviewed by: N. Karapetiants
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 47B37
Mathematics Subject Classification ID: 47A66
zbMATH DE Number: 1437097
zbMATH Keywords: martingle BMO-version, triangular projection, non-commutative BMO
MaRDI profile type: Publication
full work available at URL: pl00000415
OpenAlex ID: W2031934858
description: scientific article; zbMATH DE number 1437097