Noncommutative BMO spaces, coherent nuclearity, and bounded extensions of matrices (Q1594185)

Let \(\ell_2\) be the Hilbert space of all complex sequences \((x_j)^\infty_{j=-\infty}\) and let \(B(\ell_2)\) be the algebra of all bounded linear operators in \(\ell_2\). Any element \(A\in B(\ell_2)\) can be identified with its representing matrix \((a_{ij})^\infty_{i,j= -\infty}\) with respect to the standard basis of \(\ell_2\); the linear space of all complex matrices \(m= (m_{ij})^\infty_{i,j= -\infty}\) is denoted \({\mathcal M}\). It is well known that the triangular truncation operator \({\mathcal P}_+:{\mathcal M}\to {\mathcal M}\) given by \[ ({\mathcal P}_+m)_{ij}= \begin{cases} m_{ij},\quad &\text{if }i>j\\ 0,\quad &\text{if }i\leq j\end{cases} \] is not bounded on \(B(\ell_2)\) and that the operator \({\mathcal H}:= 2{\mathcal P}_+-\text{Id}\) has a certain resemblance to the familiar Hilbert transform. As an analogue of the classical VMO space, the author defines noncommutative VMO as the sum \(B(\ell_2)+{\mathcal H}(B(\ell_2))\) and characterizes this space as follows. Theorem 2. \(U\in B(\ell_2)+{\mathcal H}(B(\ell_2))\) if and only if \(\sup_{k\in\mathbb{Z}}\|(\text{Id}- E_k) UE_{k+1}\|_{B(\ell_2)}< \infty\) and \[ \sup_{k\in\mathbb{Z}} \|(\text{Id}- E_k) U^* E_{k+1}\|_{B(\ell_2)}< \infty, \] where \(E_k\) is orthogonal projection from \(\ell_2\) onto \(H_k:= \{(x_j)^\infty_{j=-\infty}: x_j= 0, j> k\}\). The author also outlines the connection between this description and that of coherently nuclear operators in couples of Hilbert spaces [see also the author, Mat. Zametki 63, No. 6, 866-872 (1998; Zbl 0923.47013)].