an:07166403
Zbl 1443.46035
Jana, Nabin K.; Karn, Anil K.; Peralta, Antonio M.
Absolutely compatible pairs in a von Neumann algebra
EN
Electron. J. Linear Algebra 35, 599-618 (2019).
00432924
2019
j
46L10 46B40 46L05
absolute compatibility; commutativity; \(C^\ast \)-algebra; von Neumann algebra; projection; partial isometry; linear absolutely compatible preservers
Summary: Let \(a\), \(b\) be elements in a unital \(C^\ast \)-algebra with \(0 \leq a, b \leq I\). The element \(a\) is absolutely compatible with \(b\) if \[|a-b| + |I-a-b| = I.\] In this note, some technical characterizations of absolutely compatible pairs in an arbitrary von Neumann algebra are found. These characterizations are applied to measure how far are two absolute compatible positive elements in the closed unit ball from being mutually orthogonal or commuting. In the case of 2 by 2 matrices, the results admit a geometric interpretation. Namely, non-commutative matrices of the form \(a = \begin{pmatrix} t & \alpha\\ \overline{\alpha} & 1-t \end{pmatrix}\) and \(b = \begin{pmatrix} x & \beta\\ \overline{\beta} & 1-x \end{pmatrix}\) with \(x, t \in (0, 1)\{\frac{1}{2}\}\), \( |\alpha|^2 < t(1-t) \) and \( |\beta|^2 < x(1-x)\), are absolutely compatible if, and only if, the corresponding point \(\widetilde{b} = (x, \Re(\beta), \Im(\beta))\) in \(\mathbb{R}^3\) lies in the ellipsoid \[\mathcal{E}a = \{\overline{x} \in \mathbb{R}^3 : d_2(\overline{x}, \widetilde{a}) + d_2(\overline{x}, \widetilde{a'}) = 1\},\] where \(d_2\) denotes the Euclidean distance in \(\mathbb{R}^3\), and the elements \(\widetilde{a}\) and \(\widetilde{a'}\), are \((t, \Re(\alpha), \Im(\alpha))\) and \((1-t, -\Re(\alpha), -\Im(\alpha))\), respectively. The description of absolutely compatible pairs of positive 2 by 2 matrices is applied to determine absolutely compatible pairs of positive elements in the closed unit ball of \(\mathbb{M}_n\).