On Automatic Continuity of 3-Homomorphisms on Banach Algebras

Constructions preserving $n$-weak amenability of Banach algebras

A surjective bounded homomorphism fails to preserve $n$-weak amenability, in general. We however show that it preserves the property if the involved homomorphism enjoys a right inverse. We examine this fact for certain homomorphisms on several Banach algebras.

Triviality of the generalized Lau product associated to a Banach algebra homomorphism

Several papers have, as their raison d'etre, the exploration of the \emph{generalized Lau product} associated to a homomorphism $T:B\to A$ of Banach algebras. In this short note, we demonstrate that the generalized Lau product is isomorphic as a Banach algebra to the usual direct product $A\oplus B$. We also correct some misleading claims made about the relationship between this generalized Lau product, and an older construction of Monfared (Studia Mathematica, 2007).

关于局部正则Banach代数上导出的连续性问题

On the continuity of derivations over locally regular Banach algebras

研究局部正则Banach代数上导出的连续性问题，证明在特定条件下L^p交叉积上的导出连续性。

Norm-attaining lattice homomorphisms

In this paper we study the structure of the set $\mbox{Hom}(X,\mathbb{R})$ of all lattice homomorphisms from a Banach lattice $X$ into $\mathbb{R}$. Using the relation among lattice homomorphisms and disjoint families, we prove that the topological dual of the free Banach lattice $FBL(A)$ generated by a set $A$ contains a disjoint family of cardinality $2^{|A|}$, answering a question of B. de Pagter and A.W. Wickstead. We also deal with norm-attaining lattice homomorphisms. For classical Banach lattices, as $c_0$, $L_p$-, and $C(K)$-spaces, every lattice homomorphism on it attains its norm, wh…

N-multipliers and their relations with n-homomorphisms

Let $A$ be a Banach algebra and $X$ be a Banach $A$-bimodule. We introduce and study the notions of $n$-multipliers and approximate local $n$-multipliers by generalizing the classical concept of multipliers from $A$ into $X$. As an algebraic result, we construct a Banach algebra consisting of $n$-multipliers on $A$ and under some mild conditions, we give a nice relation of this algebra with $n$-homomorphisms from $A$ into $\mathbb{C}$.

On uniform continuity of convex bodies with respect to measures in Banach spaces

Let $μ$ be a probability measure on a separable Banach space $X$. A subset $U\subset X$ is $μ$-continuous if $μ(\partial U)=0$. In the paper the $μ$-continuity and uniform $μ$-continuity of convex bodies in $X$, especially of balls and half-spaces, is considered. The $μ$-continuity is interesting for study of the Glivenko-Cantelli theorem in Banach spaces. Answer to a question of F. Topsøe is given.

A Godefroy-Kalton principle for free Banach lattices

Motivated by the Lipschitz-lifting property of Banach spaces introduced by Godefroy and Kalton, we consider the lattice-lifting property, which is an analogous notion within the category of Banach lattices and lattice homomorphisms. Namely, a Banach lattice $X$ satisfies the lattice-lifting property if every lattice homomorphism to $X$ having a bounded linear right-inverse must have a lattice homomorphism right-inverse. In terms of free Banach lattices, this can be rephrased into the following question: which Banach lattices embed into the free Banach lattice which they generate as a lattice-c…

Characterization of mixed n-Jordan homomorphisms and pseudo n-Jordan homomorphisms

In this article, a new notion of $n$-Jordan homomorphism namely the mixed $n$-Jordan homomorphism is introduced. It is proved that how a mixed $(n+1)$-Jordan homomorphism can be a mixed $n$-Jordan homomorphism and vice versa. By means of some examples, it is shown that the mixed $n$-Jordan homomorphisma are different from the $n$-Jordan homomorphisms and the pseudo $n$-Jordan homomorphisms. As a consequence, it shown that every mixed Jordan homomorphism from Banach algebra $\mathcal{A}$ into commutative semisimple Banach algebra $\mathcal{B}$ is automatically continuous. Under some mild condit…

On automatic continuity of the derivations on module extension of Banach algebras

In this article we shall focus on the derivations on module extension of Banach algebras and determine the general structure of them. Then we obtain some results concerning the automatic continuity of these mappings.

Cesáro sums and algebra homomorphisms of bounded operators

Let $X$ be a complex Banach space. The connection between algebra homomorphisms defined on subalgebras of the Banach algebra $\ell^{1}(\mathbb{N}_0)$ and the algebraic structure of Cesàro sums of a linear operator $T\in \mathcal{B}(X)$ is established. In particular, we show that every $(C, α)$-bounded operator $T$ induces - and is in fact characterized - by such an algebra homomorphism. Our method is based on some sequence kernels, Weyl fractional difference calculus and convolution Banach algebras that are introduced and deeply examined. To illustrate our results, improvements to bounds for A…

Perturbations of surjective homomorphisms between algebras of operators on Banach spaces

A remarkable result of Molnár [Proc. Amer. Math. Soc., 126 (1998), 853-861] states that automorphisms of the algebra of operators acting on a separable Hilbert space is stable under "small" perturbations. More precisely, if $ϕ,ψ$ are endomorphisms of $\mathcal{B}(\mathcal{H})$ such that $\|ϕ(A)-ψ(A)\|<\|A\|$ and $ψ$ is surjective then so is $ϕ$. The aim of this paper is to extend this result to a larger class of Banach spaces including $\ell_p$ and $L_p$ spaces ($1<p<+\infty$). En route to the proof we show that for any Banach space $X$ from the above class all faithful, unital, separable, ref…

Continuity and Bargmann mapping properties of quasi-Banach Orlicz modulation spaces

We deduce continuity, compactness and invariance properties for quasi-Banach Orlicz modulation spaces. We characterize such spaces in terms of Gabor expansions and by their images under the Bargmann transform.

Automatic continuity of Some linear mappings from certain products of Banach algebras

Let $\mathcal{A}$ and $\mathcal{U}$ be Banach algebras and $θ$ be a nonzero character on $\mathcal{A}$. Then the \textit{Lau product Banach algebra} $\mathcal{A}\times_θ\mathcal{U}$ associated with the Banach algebras $\mathcal{A}$ and $\mathcal{U}$ is the $l^1$-direct sum $\mathcal{A}\oplus\mathcal{U}$ equipped with the algebra multiplication $(a,u)(a',u')=(ab,θ(a)u'+θ(a')u+uu')\, (a,a'\in\mathcal{A},u,u'\in\mathcal{U})$ and $l^1$-norm. In this paper we shall investigate the derivations and multipliers from this Banach algebras and study the automatic continuity of these mappings. We also stu…

Free and Projective Banach Lattices

We define and prove the existence of free Banach lattices in the category of Banach lattices and contractive lattice homomorphisms and establish some of their fundamental properties. We give much more detailed results about their structure in the case that there are only a finite number of generators and give several Banach lattice characterizations of the number of generators being, respectively, one, finite or countable. We define a Banach lattice $P$ to be projective if whenever $X$ is a Banach lattice, $J$ a closed ideal in $X$, $Q:X\to X/J$ the quotient map, $T:P\to X/J$ a linear lattice …

When are full representations of algebras of operators on Banach spaces automatically faithful?

We examine the phenomenon when surjective algebra homomorphisms between algebras of operators on Banach spaces are automatically injective. In the first part of the paper we shall show that for certain Banach spaces $X$ the following property holds: For every non-zero Banach space $Y$ every surjective algebra homomorphism $ψ: \, \mathcal{B}(X) \rightarrow \mathcal{B}(Y)$ is automatically injective. In the second part of the paper we consider the question in the opposite direction: Building on the work of Kania, Koszmider and Laustsen \textit{(Trans. London Math. Soc., 2014)} we show that for e…

A remark on continuity of positive linear functionals on separable Banach *-algebras

Using a variation of the Murphy-Varopoulos Theorem, we give a new proof of the following R.J.Loy Theorem: Let A be a separable Banach*-algebra with center Z such that ZA has at most countable codimension, then every positive linear functional on A is continuous.

n-Homomorphisms

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.

Compact homomorphisms between algebras of $C(K)$-valued Lipschitz functions

We give a complete description of homomorphisms between two Banach algebras of Lipschitz functions with values in continuous functions. We also characterize the compactness of those homomorphisms.

Compact failure of multiplicativity for linear maps between Banach algebras

We introduce notions of compactness and weak compactness for multilinear maps from a product of normed spaces to a normed space, and prove some general results about these notions. We then consider linear maps $T:A\to B$ between Banach algebras that are "close to multiplicative" in the following senses: the failure of multiplicativity, defined by $S_T(a,b)=T(a)T(b)-T(ab)$, is compact [respectively weakly compact]. We call such maps cf-homomorphisms [respectively wcf-homomorphisms]. We also introduce a number of other, related definitions. We state and prove some general theorems about these ma…

Approximate Homomorphisms of Ternary Semigroups

A mapping $f:(G_1,[ ]_1)\to (G_2,[ ]_2)$ between ternary semigroups will be called a ternary homomorphism if $f([xyz]_1)=[f(x)f(y)f(z)]_2$. In this paper, we prove the generalized Hyers--Ulam--Rassias stability of mappings of commutative semigroups into Banach spaces. In addition, we establish the superstability of ternary homomorphisms into Banach algebras endowed with multiplicative norms.

Banach空间上线性余切Oseledets子空间的Hölder连续性

Hölder continuity of Oseledets subspaces for linear cocycles on Banach spaces

本文研究了Banach空间上线性余切的Oseledets子空间，在可逆与不可逆情形下证明了其Hölder连续性，推广了已有结果。

The SHAI property for the operators on L^p

A Banach space X has the SHAI (surjective homomorphisms are injective) property provided that for every Banach space Y, every continuous surjective algebra homomorphism from the bounded linear operators on X onto the bounded linear operators on Y is injective. The main result gives a sufficient condition for X to have the SHAI property. The condition is satisfied for L^p (0, 1) for 1 < p < \infty, spaces with symmetric bases that have finite cotype, and the Schatten p-spaces for 1 < p < \infty.

Continuity of coordinate functionals of filter bases in Banach spaces

We prove that the coordinate functionals associated with filter bases in Banach spaces are continuous as long as the underlying filter is analytic. This removes the large-cardinal hypothesis from the result established by the two last-named authors ([Bull. Lond. Math. Soc. 53 (2021)]) at the expense of reducing the generality from projective to analytic. In particular, we obtain a ZFC solution to Kadets' problem of continuity of coordinate functionals associated with bases with respect to the filter of statistical convergence. Even though the automatic continuity of coordinate functionals beyo…

半射性巴拿赫格局

Semiprojective Banach lattices

研究半射性巴拿赫格局的性质，证明Banach格C(X)半射性当且仅当X是绝对邻域拉回，且展示无限可数的ℓ₁和的1+项目性巴拿赫格局为半射性但不一定是1+项目性，同时证明ℓ_p和L_p空间不满足半射性。

Large cardinals and continuity of coordinate functionals of filter bases in Banach spaces

Assuming the existence of certain large cardinal numbers, we prove that for every projective filter $\mathscr F$ over the set of natural numbers, $\mathscr{F}$-bases in Banach spaces have continuous coordinate functionals. In particular, this applies to the filter of statistical convergence, thereby we solve a problem by V. Kadets (at least under the presence of certain large cardinals). In this setting, we recover also a result of Kochanek who proved continuity of coordinate functionals for countably generated filters (Studia Math., 2012).

Multigraph limits, unbounded kernels, and Banach space decorated graphs

We present a construction that allows us to define a limit object of Banach space decorated graph sequences in a generalized homomorphism density sense. This general functional analytic framework provides a universal language for various combinatorial limit notions. In particular it makes it possible to assign limit objects to multigraph sequences that are convergent in the sense of node-and-edge homomorphism numbers, and it generalizes the limit theory for graph sequences with compact decorations.

Module extensions of dual Banach algebras

In this paper we define the module extension dual Banach algebras and we use this Banach algebras to finding the relationship between $weak^*-$continuous homomorphisms of dual Banach algebras and Connes-amenability. So we study the $weak^*-$continuous derivations on module extension dual Banach algebras.

非阿基米德巴拿赫代数微分算子的连续性

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。

Effective descent morphisms for Banach modules

It is proved that a norm-decreasing homomorphism of commutative Banach algebras is an effective descent morphism for Banach modules if and only if it is a weak retract.

On a conjecture concerning some automatic continuity theorems

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.

Continuity and Schatten properties for pseudo-differential operators with symbols in quasi-Banach modulation spaces or Hörmander classes

We establish continuity and Schatten-von Neumann properties for matrix operators with matrices satisfying mixed quasi-norm estimates. These considerations also include the case when the Lebesgue and Schatten parameters are allowed to stay between $0$ and $1$. We use the results to deduce continuity and Schatten-von Neumann properties for pseudo-differential operators with symbols in a broad class of modulation spaces.

Homomorphisms on algebras of analytic functions on non-symmetrically regular spaces

We study homomorphisms on the algebra of analytic functions of bounded type on a Banach space. When the domain space lacks symmetric regularity, we show that in every fiber of the spectrum there are evaluations (in higher duals) which do not coincide with evaluations in the second dual. We also consider the commutativity of convolutions between evaluations. We show that in some Banach spaces $X$ (for example, $X=\ell_1$) the only evaluations that commute with every other evaluation in $X''$ are those in $X$. Finally, we establish conditions ensuring the symmetry of the canonical extension of a…

Invertibility preserving linear maps on the semi-simple Banach algebras

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 ...

On continuity of measurable group representations and homomorphisms

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, …

n-Jordan homomorphisms

Let $n\in \Bbb N,$ and let $A,B$ be two rings. An additive map $h: A\to B$ is called n-Jordan homomorphism if $h(a^n)=(h(a))^n$ for all $a \in {A}$. Every Jordan homomorphism is an n-Jordan homomorphism, for all $n\geq 2,$ but the converse is false, in general. In this paper we investigate the n-Jordan homomorphisms on Banach algebras. Indeed some results related to continuity are given as well.

$Δ$-weak character amenability of certain Banach algebras

In this paper we introduce the notion of $Δ$-weak character amenable Banach algebras and investigate $Δ$-weak character amenability of certain Banach algebras such as projective tensor product $A\widehat{\otimes}B$, Lau product $A\times_TB$ for every Banach algebra $ A$ and $B$, where $T$ be a homomorphism from $B$ into $A$, abstract Segal algebras and module extension Banach algebras.

A bicommutant theorem for dual Banach algebras

A dual Banach algebra is a Banach algebra which is a dual space, with the multiplication being separately weak$^*$-continuous. We show that given a unital dual Banach algebra $\mc A$, we can find a reflexive Banach space $E$, and an isometric, weak$^*$-weak$^*$-continuous homomorphism $π:\mc A\to\mc B(E)$ such that $π(\mc A)$ equals its own bicommutant.

Continuous functionals for unbounded convergence in Banach lattices

Recently, the different types of unbounded convergences (uo, un, uaw, uaw*) in Banach lattices were studied. In this paper, we study the continuous functionals with respect to unbounded convergences. We first characterize the continuity of linear functionals for these convergences. Then we define the corresponding unbounded dual spaces and get their exact form. Based on these results, we discuss order continuity and reflexivity of Banach lattices. Some related results are obtained as well.

The Complete Continuity Property and Finite Dimensional Decompositions

A Banach space $\X$ has the complete continuity property (CCP) if each bounded linear operator from $L_1$ into $\X$ is completely continuous (i.e., maps weakly convergent sequences to norm convergent sequences). The main theorem shows that a Banach space failing the CCP (resp., failing the CCP and failing cotype) has a subspace with a finite dimensional decomposition (resp., basis) which fails the CCP.

On norm continuity, differentiability and compactness of perturbed semigroups

The main purpose of this paper is to treat semigroups properties, like norm continuity, compactness and differentiability for perturbed semigroups in Banach spaces. In particular, we investigate three large classes of perturbations, Miyadera-Voigt, Desch-Schappacher and Staffans-Weiss perturbations. Our approach is mainly based on feedback theory of Salamon-Weiss systems. Our results are applied to abstract boundary integro-differential equations in Banach spaces.

Automatic continuity for the unitary group

We show that every homomorphism from the infinite-dimensional unitary or orthogonal group to a separable group is continuous.

Banach代数中的交换保持映射

Commutativity preserving mappings in Banach algebras

研究Banach代数中保持交换性的映射，证明在特定条件下，此类映射可分解为同态与反同态的和，结合中心元和加法映射。

Stability of the Jensen--type functional equation in ternary Banach algebras: An alternative fixed point approach

Using fixed point methods, we prove the generalized Hyers--Ulam--Rassias stability of ternary homomorphisms, and ternary multipliers in ternary Banach algebras for the Jensen--type functional equation $$f(\frac{x+y+z}{3})+f(\frac{x-2y+z}{3})+f(\frac{x+y-2z}{3})= f(x).$$

Maps preserving two-sided zero products on Banach algebras

Let $A$ and $B$ be Banach algebras with bounded approximate identities and let $Φ:A\to B$ be a surjective continuous linear map which preserves two-sided zero products (i.e., $Φ(a)Φ(b)=Φ(b)Φ(a)=0$ whenever $ab=ba=0$). We show that $Φ$ is a weighted Jordan homomorphism provided that $A$ is zero product determined and weakly amenable. These conditions are in particular fulfilled when $A$ is the group algebra $L^1(G)$ with $G$ any locally compact group. We also study a more general type of continuous linear maps $Φ:A\to B$ that satisfy $Φ(a)Φ(b)+Φ(b)Φ(a)=0$ whenever $ab=ba=0$. We show in particul…

Biflat-like Banach algebras

Given a Banach algebra $ \mathcal{A} $ and a continuous homomorphism $σ$ on it, the notion of $σ$-biflatness for $ \mathcal{A} $ is introduced. This is a generalization of biflatness and it is shown that they are distinct. The relations between $σ$-biflatness and some other close concepts such as $σ$-biprojectivity and $σ$-amenability are studied. The $σ$-biflatness of tensor product of Banach algebras are also discussed.

Derivations on semidirect products of Banach algebras

Let A and U be Banach algebras such that U is also a Banach A- bimodule with compatible algebra operations, module actions and norm. By defining an approprite action, we turn l1-direct product A item U into a Banach algebra such that A is closed subalgebra and U is a closed ideal of it. This algebra, is in fact semidirect product of A and U which we denote it by A litem U and every semidirect products of Banach algebras can be represented as this form. In this paper we consider the Banach algebra A litem U as mentioned and study the derivations on it. In fact we consider the automatic continui…

Slice continuity for operators and the Daugavet property for bilinear maps

We introduce and analyse the notion of slice continuity between operators on Banach spaces in the setting of the Daugavet property. It is shown that under the slice continuity assumption the Daugavet equation holds for weakly compact operators. As an application we define and characterise the Daugavet property for bilinear maps, and we prove that this allows us to describe some $p$-convexifications of the Daugavet equation for operators on Banach function spaces that have recently been introduced.

不存在通用可分巴拿赫代数

There is no universal separable Banach algebra

该研究证明不存在对所有可分巴拿赫代数同态嵌入通用的可分巴拿赫代数，在交换范畴也成立，通过构造$A_X$并结合Johnson和Szankowski的定理完成简短直接的证明。

Character amenability and character inner amenability of morphism product of Banach algebras

Let $T$ be a Banach algebra homomorphism from a Banach algebra $\mathcal B$ to a Banach algebra $\mathcal A$ with $\|T\|\leq 1$. Recently it has been obtained some results about Arens regularity and also various notions of amenability of $\mathcal A\times_{T}\mathcal B$, in the case where $\mathcal A$ is commutative. In the present paper, most of these results have been generalized and proved for an arbitrary Banach algebra $\mathcal A$.

广义卷积代数上缠绕算子的连续性研究

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。

Duality and Operator Algebras II: Operator Algebras as Banach Algebras

We answer, by counterexample, several open questions concerning algebras of operators on a Hilbert space. The answers add further weight to the thesis that, for many purposes, such algebras ought to be studied in the framework of operator spaces, as opposed to that of Banach spaces and Banach algebras. In particular, the `nonselfadjoint analogue' of a W*-algebra resides naturally in the category of dual operator spaces, as opposed to dual Banach spaces. We also show that an automatic w*-continuity result in the preceding paper of the authors is sharp.

Vector-valued characters on Vector-valued Function Algebras

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…

由Banach空间诱导的正齐函数Banach空间

Banach lattices of positively homogeneous functions induced by a Banach space

本文研究了由Banach空间诱导的正齐函数Banach空间，探讨了其格同态性质及空间之间的关系，揭示了Banach空间有限维和反射性的特征化条件。

Zero product determined Banach algebras

Let $\mathcal{L}$ be a completely distributive commutative subspace lattice or a subspace lattice with two atoms, we use a unified approach to study the derivations, homomorphisms on $\mathrm{Alg} \mathcal{L}$. We verify that the multiplier algebra of $\mathrm{Alg} \mathcal{L}\cap \mathcal{K}(\mathcal{H})$ is isomorphic to $\mathrm{Alg} \mathcal{L}$ and $\mathrm{Alg} \mathcal{L}$ is zero product determined. For $T$ in $M_{n}(\mathbb{C})$, $n\geq 2$, we show that $\mathcal{A}_{T}$ is zero product determined if and only if every local derivation from $\mathcal{A}_{T}$ into any Banach $\mathcal{A…