Modularity in the Lattice of Projections of a von Neumann Algebra

Canadian journal of mathematics, Tome 36 (1984) no. 6, pp. 1021-1030

Résumé

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.

Détail

DOI : 10.4153/CJM-1984-058-6

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