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Absolutely compatible pair of elements in a von Neumann algebra. II. (English) Zbl 1448.46048
The author considers absolutely compatible pairs of elements in a von Neumann algebra \(\mathcal M\) on a complex Hilbert space \(\mathcal H\). Denote by \(\mathcal P(\mathcal M)\) the set of all projections in \(\mathcal M\). Let \(a\in\mathcal M\) with \(0\leq a\leq I\). Put \(s(a)=\sup\{p\in\mathcal P(\mathcal M):p\leq a\}\), \(n(a)=\sup\{p\in\mathcal P(\mathcal M):pa=0\}\), and \(r(a)=\inf\{p\in\mathcal P(\mathcal M):a\leq \|a\|p\}\), respectively. We say that \(a\) is strict in \(\mathcal M\) if \(s(a) =0\) and \(n(a)=0\). A pair of elements \(0 \leq a, b \leq I\) in \(\mathcal M\) is said to be absolutely compatible if \(|a-b|+|I-a-b| = I\).
The author firstly gives a very nice sufficient condition for a pair \(0\leq A,B\leq I\) in \(M_2(\mathcal M)\) to be absolutely compatible. Let \(0\leq a,b\leq I\) in \(\mathcal M\) be a strict and commuting pair such that \(a^2+b^2\leq 1\) with \(a^2+b^2\) strict. Put \[A=\left(\begin{array}{ccc} a^2& ab \\ab &1-a^2 \\ \end{array} \right) \quad \text{and}\quad B=\left( \begin{array}{ccc} b^2& -ab \\-ab& 1-b^2 \\ \end{array} \right).\] Then \( A \) and \(B\) are absolutely compatible in \(M_2(\mathcal M)\). Moreover, a complete description of an absolutely compatible pair of strict elements in \(\mathcal M\) is given. Let \(0 \leq a, b \leq I\) in \(\mathcal M\) be strict and absolutely compatible. Put \(p=I-r\left(\frac12(ab+ba)\right)\). Then \(\mathcal H\) is unitarily equivalent to \( p(\mathcal H)\oplus p(\mathcal H)\), and there exist strict elements \(a_1,b_1\) in \(p\mathcal Mp\) with \(a_1b_1=b_1a_1\), \(a_1+b_1\leq p\) together with \(a_1+b_1\) strict in \(p\mathcal Mp\) and a unitary operator \(U\) from \(\mathcal H\) onto \(p(\mathcal H)\oplus p(\mathcal H)\) such that \[a=U^*\left( \begin{array}{ccc} a_1& (a_1b_1)^{\frac12} \\(a_1b_1)^{\frac12} &p-a_1 \\ \end{array} \right) U \quad\text{and}\quad b=U^*\left( \begin{array}{ccc} b_1& -(a_1b_1)^{\frac12} \\-(a_1b_1)^{\frac12}& p-b_1 \\ \end{array} \right)U.\]
For Part I, see [N. K. Jana et al., Electron. J. Linear Algebra 35, 599–618 (2019; Zbl 1443.46035)].
Reviewer: Guoxing Ji (Xian)
MSC:
46L10 General theory of von Neumann algebras
46B40 Ordered normed spaces
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