Schröder Bernstein Theorems for Operator Algebras
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High Quality Content by WIKIPEDIA articles! The Schröder Bernstein theorem, from set theory, has analogs in the context operator algebras. This article discusses such operator-algebraic results. Suppose M is a von Neumann algebra and E, F are projections in M. Let ~ denote the Murray-von Neumann equivalence relation on M. Define a partial order " on the family of projections by E " F if E ~ F' F. In other words, E " F if there exists a partial isometry U M such that U U = E and UU F. Suppose M is a von Neumann algebra and E, F are projections in M. Let ~ denote the Murray-von Neumann equivale...
High Quality Content by WIKIPEDIA articles! The Schröder Bernstein theorem, from set theory, has analogs in the context operator algebras. This article discusses such operator-algebraic results. Suppose M is a von Neumann algebra and E, F are projections in M. Let ~ denote the Murray-von Neumann equivalence relation on M. Define a partial order " on the family of projections by E " F if E ~ F' F. In other words, E " F if there exists a partial isometry U M such that U U = E and UU F. Suppose M is a von Neumann algebra and E, F are projections in M. Let ~ denote the Murray-von Neumann equivalence relation on M. Define a partial order " on the family of projections by E " F if E ~ F' F. In other words, E " F if there exists a partial isometry U M such that U U = E and UU F. For closed subspaces M and N where projections PM and PN, onto M and N respectively, are elements of M, M " N if PM " PN.
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