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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 Full Text: References:  [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
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