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Absolutely compatible pairs in a von Neumann algebra. (English) Zbl 1443.46035
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 \(\widetilde{b} = (x, \operatorname{Re}(\beta), \operatorname{Im}(\beta))\) 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 \((t, \operatorname{Re}(\alpha), \operatorname{Im}(\alpha))\) and \((1-t, -\operatorname{Re}(\alpha), -\operatorname{Im}(\alpha))\), 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 \(\mathbb{M}_n\).

46L10 General theory of von Neumann algebras
46B40 Ordered normed spaces
46L05 General theory of \(C^*\)-algebras
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