Hung Le Pham
We give a description of the continuity ideals and the kernels of homomorphisms from the algebras of continuous functions on locally compact spaces into Banach algebras.
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A linear map $ϕ:{\mathcal A}\to {\mathcal B} $ between (Banach) algebras is called 3-homomorphism if $ϕ(abc)=ϕ(a)ϕ(b)ϕ(c)$ for each $a, b, c \in {\mathcal A}$. We investigate 3-homomorphisms on Banach algebras with bounded approximate identities and establish in two ways that every involution preserving homomorphism between $C^*$-algebras is norm decreasing.
Let A and B be commutative locally convex algebras with unit. A is assumed to be a uniform topological algebra. Let h be an injective homomorphism from A to B. Under additional assumptions, we characterize the continuity of the homomorphism h^(-1) / Im(h) by the fact that the radical (or strong radical) of the closure of Im(h) has only zero as a common point with Im(h). This gives an answer to a conjecture concerning some automatic continuity theorems on uniform topological algebras.
Let G be a locally compact group, and let U be its unitary representation on a Hilbert space H. Endow the space L(H) of linear bounded operators on H with weak operator topology. We prove that if U is a measurable map from G to L(H) then it is continuous. This result was known before for separable H. To prove this, we generalize a known theorem on nonmeasuralbe unions of point finite families of null sets. We prove also that the following statement is consistent with ZFC: every measurable homomorphism from a locally compact group into any topological group is continuous. This relies, in turn, …
The main purpose of this article is to initiate a systematic study of Semihypergroups, first introduced by C. Dunkl [4], I. Jewett [13] and R. Spector [20] independently around 1972. We introduce and study several natural algebraic and analytic structures on semihypergroups, which are well-known in the case of topological groups and semigroups. In particular, we first study almost periodic and weakly almost periodic function spaces (basic properties, their relation to the compactness of the underlying space, introversion and Arens product on their duals among others). We then introduce homomor…
We show that every homomorphism from the infinite-dimensional unitary or orthogonal group to a separable group is continuous.
Let $\mathcal A$ and $\mathcal B$ be two (complex) algebras. A linear map $ϕ:{\mathcal A}\to{\mathcal B}$ is called $n$-homomorphism if $ϕ(a_{1}... a_{n})=ϕ(a_{1})...ϕ(a_{n})$ for each $a_{1},...,a_{n}\in{\mathcal A}.$ In this paper, we investigate $n$-homomorphisms and their relation to homomorphisms. We characterize $n$-homomorphisms in terms of homomorphisms under certain conditions. Some results related to continuity and commutativity are given as well.
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。
Entire monogenic functions of given proximate order and continuous homomorphisms
本文受量子力学中无限阶微分算子启发，引入近似阶研究整单演函数空间上的同态，刻画了其上连续作用的无限阶微分算子，为相关算子理论提供了新结果。
We give a survey of Husain, Ng and Liang's results concerning the automatic continuity of algebra homomorphisms. We also give the improvements obtained by Joseph, Akkar, Oudadess, Zelazko and ourselves.
We study automatic injectivity of surjective algebra homomorphisms from $\mathscr{B}(X)$, the algebra of (bounded, linear) operators on $X$, to $\mathscr{B}(Y)$, where $X$ is one of the following \emph{long} sequence spaces: $c_0(λ)$, $\ell_{\infty}^c(λ)$, and $\ell_p(λ)$ ($1 \leqslant p < \infty$) and $Y$ is arbitrary. \textit{En route} to the proof that these spaces do indeed enjoy such a property, we classify two-sided ideals of the algebra of operators of any of the aforementioned Banach spaces that are closed with respect to the `sequential strong operator topology'.
Derivations and homomorphisms in commutator-simple algebras
该研究提出换位子单代数的概念，刻画其导子的判定条件并应用于局部导子，证明幺模局部紧群上的$L^1(G)$连续局部导子均为导子，还讨论了该类代数中的类同态映射。
The paper describes homomorphisms between Douglas algebras and some semisimple Banach algebras. The main tool is a result on the structure of the space $C(Z,\mathfrak M)$ of continuous mappings from a connected first-countable $T_1$ space $Z$ to the maximal ideal space $\mathfrak M$ of the algebra $H^\infty$ of bounded holomorphic functions on the unit disk $\mathbb D\subset\mathbb C$. In particular, it is shown that the space of continuous mappings from $Z$ to $\mathbb D$ is dense in the topology of pointwise convergence in $C(Z,\mathfrak M)$ and the homotopy groups of $\mathfrak M$ are trivi…
In this paper we study the ideal amenability of Banach algebras. Let $\cal A$ be a Banach algebra and let $I$ be a closed two-sided ideal in $\cal A$, $\cal A$ is $I$-weakly amenable if $H^{1}({\cal A},I^*)=\{0\}$. Further, $\cal A$ is ideally amenable if $\cal A$ is $I$-weakly amenable for every closed two-sided ideal $I$ in $\cal A$. We know that a continuous homomorphic image of an amenable Banach algebra is again amenable. We show that for ideal amenability the homomorphism property for suitable direct summands is true similar to weak amenability and we apply this result for ideal amenabil…
In this paper, we show that the essentiality of the scole of an ideal B i a semi-simple Banach algebra A implies that any invertibility preserving isomorphism on A is a Jordan homomorphism. Specially ...
Locally solid convergences and order continuity of positive operators
本文研究向量格上的局部实心收敛结构，刻画序收敛与 uo 收敛，推广无界修正结果，并建立正算子序连续性在稠密正则子格上的判定条件及反例。
Notions of convergence and continuity specifically adapted to Riesz ideals I of the space of continuous real-valued functions on a Lindelöf locally compact Hausdorff space are given, and used to prove Stone-Weierstraß-type theorems for I. As applications, sufficient conditions are discussed that guarantee that various types of positive linear maps on I are uniquely determined by their restriction to various point-separating subsets of I. A very special case of this is the characterization of the strong determinacy of moment problems, which is rederived here in a rather general setting and with…
We obtain a complete description of closed ideals in weighted Lipschitz algebras $Λ_ω$ of analytic functions on the unit disk satisfying the following condition $$\frac{|f(z)-f(w)|}{ω(|z-w|)}=o(1)\qquad(as |z-w| \longrightarrow 0),$$ where $ω$ is a modulus of continuity satisfying some regularity conditions.
We examine rigidity phenomena for representations of amenable operator algebras which have an ideal of compact operators. We establish that a generalized version of Kadison's conjecture on completely bounded homomorphisms holds for the algebra if the associated quotient is abelian. We also prove that injective completely bounded representations of the algebra are similar to complete isometries. The main motivating example for these investigations is the recent construction of Choi, Farah and Ozawa of an amenable operator algebra that is not similar to a $C^*$-algebra, and we show how it fits i…
On the continuity of intertwining operators over generalized convolution algebras
该数学研究针对广义卷积代数上的缠绕算子连续性问题，借助连续理想给出自动连续的条件，提供满足条件的代数实例，并证明特定群与自由作用下的$\ell^1_\alpha({\sf G},C(X))$到巴拿赫代数的同态均自动连续。
Continuity of differential operators for nonarchimedean Banach algebras
针对非阿基米德域$K$和交换诺特巴拿赫$K$-代数$A$，研究有限生成巴拿赫$A$-模间Grothendieck意义下$K$-线性微分算子连续性，给出$K$特征为零时算子连续的充要条件：$A$每个极大理想商是$K$的有限扩张。
Absolutely continuous commuting row contractions admit a weak-$*$ continuous functional calculus. Building on recent work describing the first and second dual spaces of the closure of the polynomial multipliers on the Drury-Arveson space, we give a complete characterization of these commuting row contractions in measure theoretic terms. We also establish that completely non-unitary row contractions are necessarily absolutely continuous, in direct parallel with the case of a single contraction. Finally, we consider refinements of this question for row contractions that are annihilated by a give…
Harmonic analysis and automatic continuity in the context of generalized differential subalgebras
本文引入并系统研究$(k,p,q)$-微分子代数类，证明其具有函数演算封闭性、$^*$-正则性、Wiener性质及自动连续性等关键性质。
We show that ideal submodules and closed ternary ideals in Hilbert modules are the same. We use this insight as a little peg on which to hang a little note about interrelations with other notions regarding Hilbert modules. In Section 3, we show that the ternary ideals (and equivalent notions) merit fully, in terms of homomorphisms and quotients, to be called ideals of (not necessarily full) Hilbert modules. The properties to be checked are intrinsically formulated for the modules (without any reference to the algebra over which they are modules) in terms of their ternary structure. The proofs,…
We present the following reflexivity-like result concerning the automorphism group of the $C^*$-algebra B(H), H being a separable Hilbert space. Let $ϕ:B(H)\to B(H)$ be a multiplicative map (no linearity or continuity is assumed) which can be approximated at every point by automorphisms of B(H) (these automorphisms may, of course, depend on the point) in the operator norm. Then $ϕ$ is an automorphism of the algebra B(H).
A note on finite-dimensional quotients and the problem of automatic continuity for twisted convolution algebras
本文证明扭曲卷积代数的有限余维闭理想商为半单代数，并借此扩展自动连续性结果，构造大量相关例子。
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