B(H) lattices, density and arithmetic mean ideals
2011; University of Houston; Volume: 37; Issue: 1 Linguagem: Inglês
ISSN
0362-1588
Autores Tópico(s)Advanced Algebra and Logic
ResumoThis paper studies lattice properties of operator ideals in B(H) and their appli- cations to the arithmetic mean ideals which were introduced in (10) and further studied in the project (14)-(17) of which this paper is a part. It is proved that the lattices of all principal ideals, of principal ideals with a generator that satisfies the �1/2-condition, of arithmetic mean stable principal ideals (i.e., those with an am-regular generator), and of arithmetic mean at infinity stable principal ideals (i.e., those with an am-∞ regular generator) are all both upper and lower dense in the lattice of general ideals. This means that between any ideal and an ideal (nested above or below respectively) in one of these sublattices, lies another ideal in that sublattice. Among the applications: a principal ideal is am-stable (and similarly for am-∞ stable principal ideals) if and only if any (or equivalently, all) of its first order arithmetic mean ideals are am-stable, such as its am-interior, am-closure and others. A principal ideal I is am-stable (and similarly for am-∞ stable principal ideals) if and only if it satisfies any (equivalently, all) of the first order equality cancellation properties, e.g, Ja = Ia ⇒ J = I. These cancellation properties can fail even for am-stable countably generated ideals. It is proven that while the inclusion cancellation Ja ⊃ Ia implies J ⊃ I does not hold in general, even for I am-stable and principal, there is always a largest ideal b I for which Ja ⊃ Ia ⇒ J ⊃ b I. Furthermore, if I = (�) is principal, then b I is principal as ' for the harmonic sequence !, 0 < p < 1, and 1/p − 1/p ' = 1.
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