Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0707.3115

B(H) lattices, density and arithmetic mean ideals

Victor Kaftal·Gary Weiss
View original source ↗Read source document ↗Export metadata ↓

Abstract

This part of a multi-paper project studies the lattice properties of the arithmetic mean ideals of B(H) introduced by Dykema, Figiel, Weiss, and Wodzicki. We prove: the lattices of all principal ideals, of arithmetic mean or arithmetic mean at infinity stable principal ideals or of principal ideals with a generator that satisfies the Delta_1/2 condition, are all both upper and lower dense in the lattice of general ideals. That is, 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 I is am-stable (I = I_a) if and only if any of its first order arithmetic mean ideals are am-stable if and only if the ideal satisfies the first order equality cancellation property: J_a = I_a implies J = I. We show that this cancellation property can fail even for am-stable countably generated ideals. Similar results hold for arithmetic mean at infinity ideals. Inclusion cancellations do not hold in general even for principal ideals, but for every ideal I there is a largest ideal I^ for which J_a contains I_a implies that J contains I^. When I is principal, I^ too is principal. We show that I=I^ is a strictly stronger property than am-stability. For example, for I the p < 1 power of the principal ideal J generated by diag {1/n}, I^ is the q power of J where 1/p-1/q = 1.

Explore related subjects

math.FAmath.OA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Victor Kaftal, Gary Weiss. 2007-07-20. B(H) lattices, density and arithmetic mean ideals. https://arxiv.org/abs/0707.3115

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Operations on $\mathscr{M}\otimes_{\mathcal{A}(U)}\mathrm{Alt}^{\bullet}(U,M)$. Stokes theorem for locally convex linear space valued forms. Divergence theorem for locally convex linear space valued vector fields

We extend to $\mathscr{M}\otimes_{\mathcal{A}(U)}\mathfrak{T}_{\bullet}^{\bullet}(U,M)$ the usual operations defined on $\mathfrak{T}_{\bullet}^{\bullet}(U,M)$, then by employing the properties of the projective tensor product of locally convex spaces, we generalize to $\mathscr{M}\otimes_{\mathcal{A}(U)}\mathrm{Alt}^{\bullet}(U,M)$ the usual wedge product, insertion operator and exterior differential defined on $\mathrm{Alt}^{\bullet}(U,M)$ and establish their properties. Here $M$ is a smooth finite dimensional manifold, $U$ an open submanifold of $M$, $\mathcal{A}(U)$ the ring of smooth maps on $U$, $\mathfrak{T}_{\bullet}^{\bullet}(U,M)$ the $\mathcal{A}(U)$-module of smooth tensor fields of $M$ defined on $U$, $\mathrm{Alt}^{\bullet}(U,M)$ the $\mathcal{A}(U)$-module of smooth alternating tensor fields of $M$ defined on $U$, while $\mathscr{M}$ is a $\mathcal{A}(U)$-module. $\mathscr{M}$ depending by the operation might be either general, or one of the following function spaces: $\mathscr{L}_{c}^{1}(U,G,λ)$, $\mathscr{L}_{loc}^{1}(U,G,λ)$, $\mathcal{B}^{p}(U,G)$ with $G$ a Hausdorff locally convex space. This framenwork permits to construct the divergence of a $G$-valued $p$-times continuously differentiable vector field, to define the weak integral of $G$-valued compactly supported scalarly integrable maximal forms, to associate a $G$-valued measure with any locally integrable $G$-valued form, to establish a Stokes type theorem for $G$-valued compactly supported $p$-times continuously differentiable forms and to obtain a divergence type theorem for $G$-valued $p$-times continously differentiable vector fields.

math.FA↗
arXiv2026

The Grothendieck Constant is Strictly Larger than Davie-Reeds' Bound

The Grothendieck constant $K_{G}$ is a fundamental quantity in functional analysis, with important connections to quantum information, combinatorial optimization, and the geometry of Banach spaces. Despite decades of study, the value of $K_{G}$ is unknown. The best known lower bound on $K_{G}$ was obtained independently by Davie and Reeds in the 1980s. In this paper we show that their bound is not optimal. We prove that $K_{G} \ge K_{DR} + 10^{-12}$, where $K_{DR}$ denotes the Davie-Reeds lower bound. Our argument is based on a perturbative analysis of the Davie-Reeds operator. We show that every near-extremizer for the Davie-Reeds problem has $Ω(1)$ weight on its degree-3 Hermite coefficients, and therefore introducing a small cubic perturbation increases the integrality gap of the operator.

math.FA↗
arXiv2026

Higher-order differentiability of Korevaar--Schoen energy forms and energy measures

In this paper, we investigate the differentiability of Korevaar--Schoen $p$-energy forms and the associated $p$-energy measures. We obtain higher-order derivatives by virtue of explicit realizations of Korevaar--Schoen $p$-energy forms and the associated $p$-energy measures as subsequential pointwise limits of certain double integrals.

math.FA↗