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Modularity in the Lattice of Projections of a von Neumann Algebra

Published online by Cambridge University Press:  20 November 2018

Donald Bures
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We say that two elements e and f of a lattice are moderately separated provided e ∧ f = 0 and both (e′, f′) and (f′, e′) are modular pairs for all e′ ≦ e and f′ ≦ f. Here (e′, f′) a modular pair means that, for all g ≧ e′,

In the lattice of projections of a factor we show that e and f, with e ∧ f = 0, are modularly separated if and only if ‖(e – k)f‖ < 1 for some finite projection k ≦ e. From there we can show that a kind of “independence property” holds for modular separation in this case: if e and f are modularly separated and if e ∨ f and g are modularly separated, then e and f ∧ g are modularly separated.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 36 , Issue 6 , 01 December 1984 , pp. 1021 - 1030
Copyright
Copyright © Canadian Mathematical Society 1984

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