Absolutely compatible pairs in a von Neumann algebra
| Autor: | Antonio M. Peralta, Anil Kumar Karn, Nabin K. Jana |
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| Rok vydání: | 2019 |
| Předmět: |
Unit sphere
Partial isometry Algebra and Number Theory Unital Mathematics - Operator Algebras Primary 46L10 Secondary 46B40 46L05 Measure (mathematics) Projection (linear algebra) Interpretation (model theory) Combinatorics symbols.namesake Von Neumann algebra FOS: Mathematics symbols Operator Algebras (math.OA) Commutative property Mathematics |
| Zdroj: | The Electronic Journal of Linear Algebra. 35:599-618 |
| ISSN: | 1081-3810 |
| DOI: | 10.13001/ela.2019.5165 |
| Popis: | Let $a,b$ be elements in a unital C$^*$-algebra with $0\leq a,b\leq 1$. The element $a$ is absolutely compatible with $b$ if $$\vert a - b \vert + \vert 1 - a - b \vert = 1.$$ In this note we find some technical characterizations of absolutely compatible pairs in an arbitrary von Neumann algebra. These characterizations are applied to measure how close is a pair of absolute compatible positive elements in the closed unit ball from being orthogonal or commutative. In the case of 2 by 2 matrices the results offer a geometric interpretation in terms of an ellipsoid determined by one of the points. The conclusions for 2 by 2 matrices are also applied to describe absolutely compatible pairs of positive elements in the closed unit ball of $\mathbb{M}_n$. 20 pages |
| Databáze: | OpenAIRE |
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