Olaf Müller
This note tries to give an answer to the following question: Is there a sufficiently rich class of metric vector spaces such that sufficiently large spaces of continuous linear maps between them are metrizable?
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We define and characterize spaces of manifold-valued generalized functions and generalized vector bundle homomorphisms in the setting of the full diffeomorphism-invariant vector-valued Colombeau algebra. Furthermore, we establish point value characterizations for these spaces.
Let $L$ be a (non necessarily unital) truncated vector lattice of real-valued functions on a nonempty set $X$. A nonzero linear functional $ψ$ on $L$ is called a truncation homomorphism if it preserves truncation, i.e.,% \[ ψ\left( f\wedge\mathbf{1}_{X}\right) =\min\left\{ ψ\left( f\right) ,1\right\} \text{ for all }f\in L. \] We prove that a linear functional $ψ$ on $L$ is a truncation homomorphism if and only if $ψ$ is a lattice homomorphism and% \[ \sup\left\{ ψ\left( f\right) :f\leq\mathbf{1}_{X}\right\} =1. \] This allows us to prove different evaluating characterizations of truncation ho…
Let $X$ and $Y$ be compact Hausdorff spaces, $E$ and $F$ be real or complex normed spaces and $A(X,E)$ be a subspace of $C(X,E)$. For a function $f\in C(X,E)$, let $\coz(f)$ be the cozero set of $f$. A pair of additive maps $S,T: A(X,E) \lo C(Y,F)$ is said to be jointly separating if $\coz(Tf)\cap \coz(Sg)=\emptyset$ whenever $\coz(f)\cap \coz(g)= \emptyset$. In this paper, first we give a partial description of additive jointly separating maps between certain spaces of vector-valued continuous functions (including spaces of vector-valued Lipschitz functions, absolutely continuous functions an…
We study the vector-valued spectrum $\mathcal{M}_{u,\infty}(B_{\ell_2},B_{\ell_2})$ which is the set of nonzero algebra homomorphisms from $\mathcal{A}_u(B_{\ell_2})$ (the algebra of uniformly continuous holomorphic functions on $B_{\ell_2}$) to $\mathcal {H}^\infty(B_{\ell_2})$ (the algebra of bounded holomorphic functions on $B_{\ell_2}$). This set is naturally projected onto the closed unit ball of $\mathcal {H}^\infty(B_{\ell_2}, \ell_2)$ giving rise to an associated fibering. Extending the classical notion of cluster sets introduced by I. J. Schark (1961) to the vector-valued spectrum we …
Lattice Homomorphisms between Sobolev Spaces
证明了索伯列夫空间之间的每个向量格同态$T$都可以用一个复合和一个乘法表示，即对于拟每/几乎每$x$和所有$u$，$T$具有$Tu(x)=u(h(x))g(x)$的形式.
Weak convergence of probability measures is one of the most important topics in the field probability and statistics. In this survey paper, we look at weak convergence of probability measures from the topological vector space point of view. We start from the key concepts and results about weak topology and weak convergence under the general framework of topological vector spaces. Then we restrict our attention to the space of probability measures and see how the general results specialize to those in probability theory. Finally, we will review some important facts about the metrizability of we…
Continuous homomorphisms of Arens-Michael algebras
证明了Arens-Michael代数的每一个连续同态都可以作为由Fréchet代数构成的某些投射系的一个态射的极限而得到.在此基础上，证明了Fréchet代数的不可数乘积的补子代数与Fréchet代数的乘积拓扑同构.这些结果被用来刻画局部凸拓扑向量空间范畴的内射对象。对偶地证明了巴拿赫空间的不可数直和的补子空间与（{ \bf LB}）-空间的直和拓扑同构.这个结果被用来描述上述类别的投射对象。
Let $A$ be a commutative Banach algebra and $X$ be a compact space. The class of Banach $A$-valued function algebras on $X$ consists of subalgebras of $C(X,A)$ with certain properties. We introduce the notion of $A$-characters on an $A$-valued function algebra $\A$ as homomorphisms from $\A$ into $A$ that basically have the same properties as the evaluation homomorphisms $\cE_x:f\mapsto f(x)$, with $x\in X$. For the so-called natural $A$-valued function algebras, such as $C(X,A)$ and $\Lip(X,A)$, we show that $\cE_x$ ($x\in X$) are the only $A$-characters. Vector-valued characters are utilized…
Simpler characterizations of total orderization invariant maps
针对分配格上全序化不变映射研究中全序化符号繁琐导致计算困难的问题，本文给出全序化映射的更简单刻画，证明格多同态全序化不变当且仅当对称，且对称格多同态的对角线是格同态，推广了向量格相关已知结果。
Aliprantis's questions on locally solid topologies
本文否定回答了Aliprantis在1974年提出的关于局部实心向量格拓扑完备化的所有问题，构造了五个反例，并在Lean 4中形式化验证。
We study the vector-valued spectrum $\mathcal{M}_\infty(B_{c_0},B_{c_0})$, that is, the set of non null algebra homomorphisms from $\mathcal H^\infty(B_{c_0})$ to $\mathcal H^\infty(B_{c_0})$ which is naturally projected onto the closed unit ball of $\mathcal H^\infty(B_{c_0}, \ell_\infty)$, likewise the scalar-valued spectrum $\mathcal M_\infty(B_{c_0})$ which is projected over $\bar{B}_{\ell_\infty}$. Our itinerary begins in the scalar-valued spectrum $\mathcal{M}_\infty(B_{c_0})$: by expanding a result by Cole, Gamelin and Johnson (1992) we prove that on each fiber there are $2^c$ disjoint …
Homomorphisms of the lattice of slowly oscillating functions on the half-line
研究半直线上慢振荡函数向量格同态空间的结构，证明零同态的特殊性，并给出该空间与半直线乘积空间加一点同胚的刻画。
We classify and explicitly describe homomorphisms of Verma modules for conformal Galilei algebras $\mathfrak{cga}_\ell(d,{\mathbb C})$ with $d=1$ for any integer value $\ell \in \mathbb{N}$. The homomorphisms are uniquely determined by singular vectors as solutions of certain differential operators of flag type, and identified with specific polynomials arising as coefficients in the expansion of a parametric family of symmetric polynomials into power sum symmetric polynomials.
For Banach spaces $X$ and $Y$ we study the vector-valued spectrum $\mathcal M_\infty(B_X,B_Y)$, that is the set of non null algebra homomorphisms from $\mathcal H^\infty(B_X)$ to $\mathcal H^\infty(B_Y)$, which is naturally projected onto the closed unit ball of $\mathcal H^\infty(B_Y, X^{**})$. The aim of this article is to describe the fibers defined by this projection, searching for analytic balls and considering Gleason parts.
Spectral theorems for positive algebra homomorphisms
本文证明了局部紧 Hausdorff 空间上正代数同态可由唯一正则谱测度表示，并将其扩展为 $L^1$-空间上的向量格同态，揭示了多种谱定理的共同序理论根源。
Order continuity and regularity on vector lattices and on lattices of continuous functions
本文刻画 Archimedean 向量格间序连续同态，借助连续函数格上的复合算子研究 $C(X)$ 子格性质，并给出正则子格刻画及 Banach 格中闭包保持正则性的结论。
Given a compact space $X$ and a commutative Banach algebra $A$, the character spaces of $A$-valued function algebras on $X$ are investigated. The class of natural $A$-valued function algebras, those whose characters can be described by means of characters of $A$ and point evaluation homomorphisms, is introduced and studied. For an admissible Banach $A$-valued function algebra $\mathcal{A}$ on $X$, conditions under which the character space $M(\mathcal{A})$ is homeomorphic to $M(\mathfrak{A}) \times M(A)$ are presented, where $\mathfrak{A}=C(X) \cap \mathcal{A}$ is the subalgebra of $\mathcal{A…
A Note on Distribution Spaces on Manifolds
本文系统地给出了n维光滑流形上向量值分布空间的26种不同的具体表示，其中大部分是新的.在特殊的情况下，表示作为模块同态作用于部分的对偶丛分别。在n-形式上，这些同态的连续性已经是它们的代数性质的结果。
Lattice homomorphisms in harmonic analysis
本文研究局部紧群闭子半群上 Banach 格间的正双线性映射，证明在温和条件下其典范关联映射为格同态，并应用于卷积问题，同时建立了序对偶的序理想刻画与 L^p 空间及测度格的嵌入结果。
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