Non-commutative BMO space
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.
martingle BMO-version, triangular projection, non-commutative BMO
scientific article; zbMATH DE number 1437097