B(h) Lattices, Density and Arithmetic Mean Ideals

This paper is part of an eight paper project [14]-[20] studying the arithmetic mean operator ideals in B(H) introduced by Dykema, Figiel, Weiss and Wodzicki in [10]. Every ideal I is generated by diagonal operators with positive decreasing sequences, and its arithmetic mean ideal Ia is generated by diagonal operators with the arithmetic means of those sequences. In this paper we focus on lattice properties: we prove that the lattices of all principal ideals, of principal ideals with ∆1/2-generators, of arithmetic mean stable principal ideals (I = Ia), and of arithmetic mean at infinity stable principal ideals (I = Ia∞ ) are all both upper and lower dense in the lattice of general ideals, i.e., between any ideal and an ideal in one of these sublattices lies another ideal in that sublattice. Among the applications: a principal ideal is am-stable (similarly for am-∞ stable) if and only if any (hence all) of its first order arithmetic mean ideals are am-stable, e.g., its am-interior, am-closure and others. A principal ideal I is am-stable (similarly for am-∞ stable) if and only if it satisfies any (hence all) of the first order equality cancelation properties, e.g., Ja = Ia ⇒ J = I. Cancelations can fail even for am-stable countably generated ideals. We prove that while the inclusion cancelation Ja ⊃ Ia ⇒ J ⊃ I does not hold in general, even for I am-stable and principal, there is always a largest ideal Î for which Ja ⊃ Ia ⇒ J ⊃ Î. Furthermore, if I is principal and has generator diag(ξ), then Î has generator diag(ξ̂) and ξ̂ is a sequence optimal in the majorization sense. That is, η ≥ ξ̂ asymptotically for all sequences η decreasing to zero for which ∑n 1 ηj ≥ ∑n 1 ξj for every n and ξ̂ is asymptotically sharp. In particular, ω̂1/p = ω1/p ′ for the harmonic sequence ω = < 1 n >, 0 < p < 1, and 1/p− 1/p′ = 1. 2000 Mathematics Subject Classification. Primary: 47B47, 47B10; Secondary: 46A45, 46B45, 47L20.