Holomorphic duality for countable discrete groups

Invariant subspaces of the integration operators on Hörmander algebras and Korenblum type spaces

We describe the proper closed invariant subspaces of the integration operator when it acts continuously on countable intersections and countable unions of weighted Banach spaces of holomorphic functions on the unit disc or the complex plane. Applications are given to the case of Korenblum type spaces and Hörmander algebras of entire functions.

Loomis--Sikorski Theorem and Stone Duality for Effect Algebras with Internal State

Recently Flaminio and Montagna, \cite{FlMo}, extended the language of MV-algebras by adding a unary operation, called a state-operator. This notion is introduced here also for effect algebras. Having it, we generalize the Loomis--Sikorski Theorem for monotone $σ$-complete effect algebras with internal state. In addition, we show that the category of divisible state-morphism effect algebras satisfying (RDP) and countable interpolation with an order determining system of states is dual to the category of Bauer simplices $Ω$ such that $\partial_e Ω$ is an F-space.

On holomorphic reflexivity conditions for complex Lie groups

We consider Akbarov's holomorphic version of the non-commutative Pontryagin duality for a complex Lie group. We prove, under the assumption that $G$ is a Stein group with finitely many components, that (1) the topological Hopf algebra of holomorphic functions on $G$ is holomorphically reflexive if and only if $G$ is linear; (2) the dual cocommutative topological Hopf algebra of exponential analytic functional on $G$ is holomorphically reflexive. We give a counterexample, which shows that the first criterion cannot be extended to the case of infinitely many components. Nevertheless, we conjectu…

Duality of holomorphic Hardy type tent spaces

We study the holomorphic tent spaces $\mathcal{HT}^p_{q,α}(\Bn)$, which are motivated by the area function description of the Hardy spaces on one hand, and the maximal function description of the Hardy spaces on the other. Characterizations for these spaces under general fractional differential operators are given. We describe the dual of $\mathcal{HT}^p_{q,α}(\Bn)$ for the full range $0<p,q<\infty$ and $α>-n-1$. Here the case $1<p<\infty$, $q\leq 1$ leads us to the Hardy-Bloch type spaces $\mathcal{BT}^{p}(\Bn)$ and $\mathcal{BT}^{p,0}(\Bn)$, the latter being the predual of $\mathcal{HT}^p_{1…

角导数密度的尖锐熵条件

A Sharp Entropy Condition For The Density Of Angular Derivatives

本文证明单位圆盘全纯自映射的有限 Carathéodory 角导数点集在熵可积条件下必为可数个有限熵 Beurling-Carleson 集之并，并构造反例说明该刻画是精确的。

Pontryagin duality for Abelian $s$- and $sb$-groups

The main goal of the article is to study the Pontryagin duality for Abelian $s$- and $sb$-groups. Let $G$ be an infinite Abelian group and $X$ be the dual group of the discrete group $G_d$. We show that a dense subgroup $H$ of $X$ is $\mathfrak{g}$-closed iff $H$ algebraically is the dual group of $G$ endowed with some maximally almost periodic $s$-topology. Every reflexive Polish Abelian group is $\mathfrak{g}$-closed in its Bohr compactification. If a $s$-topology $τ$ on a countably infinite Abelian group $G$ is generated by a countable set of convergent sequences, then the dual group of $(G…

Chain-center duality for locally compact groups

The chain group $C(G)$ of a locally compact group $G$ has one generator $g_ρ$ for each irreducible unitary $G$-representation $ρ$, a relation $g_ρ=g_{ρ'}g_{ρ"}$ whenever $ρ$ is weakly contained in $ρ'\otimes ρ"$, and $g_{ρ^*}=g_ρ^{-1}$ for the representation $ρ^*$ contragredient to $ρ$. $G$ satisfies chain-center duality if assigning to each $g_ρ$ the central character of $ρ$ is an isomorphism of $C(G)$ onto the dual $\widehat{Z(G)}$ of the center of $G$. We prove that $G$ satisfies chain-center duality if it is (a) a compact-by-abelian extension, (b) connected nilpotent, (c) countable discret…

上半格上离散全纯函数的Hardy空间

Hardy spaces of discrete holomorphic functions on the upper half-lattice

该研究在正方形格点离散全纯性框架下，发展上半格离散全纯函数的Hardy空间\(H^p\)理论，证明相关公式与恒等式，描述再生核等，比较离散与经典理论，还证明对偶性等结果。

$C^*$-Algebraic Covariant Structures

We introduce {\it covariant structures} $\left\{(\A,\k),(\a,å),\(\ha,\haa\)\right\}$ formed of a separable $C^*$-algebra $\A$, a measurable twisted action $(\a,å)$ of the second-countable locally compact group $\G$\,, a measurable twisted action $(\ha,\haa)$ of another second-countable locally compact group $\hG$ and a strictly continuous function $\k:\G\times\hG\to\U\M(\A)$ suitably connected with $(\a,å)$ and $\(\ha,\haa\)$\,. Natural notions of covariant morphisms and representations are considered, leading to a sort of twisted crossed product construction. Various $C^*$-algebras emerge by …

The character of topological groups, via bounded systems, Pontryagin--van Kampen duality and pcf theory

The Birkhoff--Kakutani Theorem asserts that a topological group is metrizable if and only if it has countable character. We develop and apply tools for the estimation of the character for a wide class of nonmetrizable topological groups. We consider abelian groups whose topology is determined by a countable cofinal family of compact sets. These are the closed subgroups of Pontryagin--van Kampen duals of \emph{metrizable} abelian groups, or equivalently, complete abelian groups whose dual is metrizable. By investigating these connections, we show that also in these cases, the character can be e…