zbMATH — the first resource for mathematics

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)
46L10 General theory of von Neumann algebras
46B40 Ordered normed spaces
Full Text: DOI
[1] Blackadar, B., Operator algebras (2006), Heidelberg Berlin: Springer-Verlag, Heidelberg Berlin
[2] Halmos, PR, Two subspaces, Trans. Am. Math. Soc., 144, 181-189 (1969)
[3] Jana, NK; Karn, AK; Peralta, AM, Absolutely compatible pairs in a von Neumann algebra, Electron J Linear Algebra, 35, 599-618 (2019) · Zbl 1443.46035
[4] Jana, NK; Karn, AK; Peralta, AM, Contractive linear preservers of absolutely compatible pairs between \(C^*\)-algebras, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales, Serie A. Matemáticas (RCSM), 113, 3, 2731-2741 (2019)
[5] Karn, AK, Orthogonality in \(C^*\)-algebras, Positivity, 20, 3, 607-620 (2016) · Zbl 1361.46017
[6] Karn, AK, Algebraic orthogonality and commuting projections in operator algebras, Acta Sci. Math. (Szeged), 84, 323-353 (2018) · Zbl 1413.46023
[7] Pedersen, GK, \(C^*\)-algebras and their automorphism groups (1979), London: Academic Press, London
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.