Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.03179

Basic Theory of Tower Systems

Quanfeng Liu·Nikica Uglešić
View original source ↗Read source document ↗Export metadata ↓

Abstract

For a Banach space $X$, we study the completed direct limit of its even iterated duals along the canonical embeddings and the bounded inverse limit of the corresponding odd duals. We give explicit constructions, establish their duality relation, and describe the comparison maps associated with iterated dualization. We then examine several assertions in three preprints of Uglešić, giving counterexamples concerning dimension stability and the commutation of bidualization with direct limits. The finite quotient shape (the category whose objects are the finite-dimensional quotients, with all continuous linear maps between them) is classified by the algebraic dimension of the continuous dual. We also compute the dimensions of the finite iterated duals and the completed tower of $\ell^{\infty}(\mathbb{N})$, and prove that, inside the tower space, the canonical copy of a higher layer need not be the range of an $M$-projection. Throughout, uncompleted limits, Banach completions, and the different meanings of isomorphism are distinguished explicitly.

Explore related subjects

math.FA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Quanfeng Liu, Nikica Uglešić. 2026-10-02. Basic Theory of Tower Systems. https://arxiv.org/abs/2610.03179

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

KEEP EXPLORING

Related papers

arXiv2026

A small Banach space $C(K)$ without nice renormings

We prove that consistently $ω_1<\mathfrak{c}$ and there exists a compact space $K$ whose Banach space $C(K)$ of continuous real-valued functions is Grothendieck, has density $ω_1$, and admits no renorming which is strictly convex or sequentially Kadets--Klee.

math.FA↗
arXiv2026

Small-2 Sets Are Riesz Sets

Let $ G $ be a compact metrizable Abelian group, $ L^{1}(G) $ its group algebra and $ M(G) $ its measure algebra. For each proper subset $ E $ of the dual group $ \hat{G} $, let $ L^{1}_{E}(G)=\{f\in L^{1}(G):\hat{f}=0 \text{ on } \hat{G}\setminus E \}$ and $M_{E}=\{μ\in M(G):\hatμ=0 \text{ on }\hat{G}\setminus E\} $. If $ M_{E}(G)=L^{1}_{E}(G) $ then the set $ E $ is said to be a Riesz sets. If $ M_{E}(G)*M_{E}(G)\subseteq L_{E}^{1}(G) $ then $ E $ is said to be a small-2 set. The main results of this paper are the following: 1. Every small-2 set is a Riesz set. 2. The ideal $ L^{1}_{E}(G) $ is Arens regular iff $ E $ is a Riesz set. Let $ A=L_{E}(G) $ and equip $ A^{**} $ with the first Arens product. 3. The centre of $ A^{**} $ is $ Z(A^{**})=A+N(A^{**}) $, where $ N(A^{**})=\{r\in A^{**}:rA^{**}=\{0\}\} $. These results settle three long-standing open problems in this area.

math.FA↗
arXiv2026

Area operators on Hardy spaces of Dirichlet series II: counterexamples and compactness criteria

We study the area operators $\mathbb{A}_{μ,l}$, $0<l<\infty$, induced by positive Borel measures on the right half-plane and acting on the Hardy spaces of Dirichlet series $\mathscr H^p$, $0<p<\infty$. We first disprove a conjecture proposed by the present authors in an earlier work by constructing a probability measure, valid for all $0<p,l<\infty$, for which the associated area operator is bounded although the measure fails the proposed Carleson conditions. We then characterize the compactness of $\mathbb{A}_{μ,l}$ on both $\mathscr H^p$ and the Hardy space $\mathscr H^p_0$ of Dirichlet series vanishing at $+\infty$, for every $0<p,l<\infty$, via integrability conditions for the associated cone integrals. In particular, boundedness and compactness are equivalent for these operators. We also give direct proofs of compactness under vanishing Carleson measure and compact $H_{\mathbb i}^p$-Carleson embedding hypotheses. The direct proofs yield estimates that can be used for measures depending on the vertical limit character $χ$. As an application, we give a different proof of a known compactness result for Volterra operators on $\mathscr H^p$ with Dirichlet series symbols in $\operatorname{VMOA}(\mathbb C_0)$.

math.FA↗