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, reflexive representations of $\mathcal B (X)$ which preserve rank one operators are in fact isomorphisms.
In this survey, we present several results related to characterizing the surjective isometries on Banach sequence spaces. Our survey includes full proofs of these characterizations for the classical spaces as well as more recent results for combinatorial Banach spaces and Tsirelson-type spaces. Along the way, we pose many open problems related to the structure of the group of surjective isometries and characterizations of the group of surjective isometries for various Banach spaces.
We characterize those classes $\mathcal{C}$ of separable Banach spaces for which there exists a separable Banach space $Y$ not containing $\ell_1$ and such that every space in the class $\mathcal{C}$ is a quotient of $Y$.
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'.
Let $H(\mathbb{D})$ be the linear space of all analytic functions on the open unit disc $\mathbb{D}$ and $H^p(\mathbb{D})$ the Hardy space on $\mathbb{D}$. The characterization of complex linear isometries on $\mathcal{S}^p=\{f\in H(\mathbb{D}):f'\in H^p(\mathbb{D})\}$ was given for $1\leq p<\infty$ by Novinger and Oberlin in 1985. Here, we characterize surjective, not necessarily linear, isometries on $\mathcal{S}^\infty$.
We show that if $T$ is an isometry (as metric spaces) between the invertible groups of unital Banach algebras, then $T$ is extended to a surjective real-linear isometry up to translation between the two Banach algebras. Furthermore if the underling algebras are closed unital standard operator algebras, $(T(e_A))^{-1}T$ is extended to a surjective real algebra isomorphism; if $T$ is a surjective isometry from the invertible group of a unital commutative Banach algebra onto that of a unital semisimple Banach algebra, then $(T(e_A))^{-1}T$ is extended to a surjective isometrical real algebra isom…
In this paper we find some necessary and sufficient conditions for a Banach algebra to be amenable or weakly amenable, by applying the homomorphisms on Banach algebras.
In this article poles, isolated spectral points, group, Drazin and Koliha-Drazin invertible elements in the context of quotient Banach algebras or in ranges of (not necessarily continuous) homomorphism between complex unital Banach algebras will be characterized using Fredholm and Riesz Banach algebra elements. Calkin algebras on Banach and Hilbert spaces will be also considered.
We construct an example of a real Banach space whose group of surjective isometries has no uniformly continuous one-parameter semigroups, but the group of surjective isometries of its dual contains infinitely many of them. Other examples concerning numerical index, hermitian operators and dissipative operators are also shown.
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 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.
We characterize the holomorphic mappings $f$ between complex Banach spaces that may be written in the form $f=T\circ g$, where $g$ is another holomorphic mapping and $T$ belongs to a closed surjective operator ideal.
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).
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…
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 $\mathbb{C}$.
Recently in a series of papers it is observed that in many Banach spaces, which include classical spaces $C(Ω)$ and $L_p$-spaces, $1 \leq p < \infty, p \neq 2$, any generalized bi-circular projection $P$ is given by $P = \frac{I+T}{2}$, where $I$ is the identity operator of the space and $T$ is a reflection, that is, $T$ is a surjective isometry with $T^2 = I$. For surjective isometries of order $n \geq 3$, the corresponding notion of projection is generalized $n$-circular projection as defined in \cite{AD}. In this paper we show that in a Banach space $X$, if generalized bi-circular projectio…
A quojection (projective limit of Banach spaces with surjective linking mappings) without infinite dimensional Banach subspaces is constructed. This results answers a question posed by G.Metafune and V.B.Moscatelli.
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…
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…
Given a separable Banach space $E$, we construct an extremely non-complex Banach space (i.e. a space satisfying that $\|Id + T^2\|=1+\|T^2\|$ for every bounded linear operator $T$ on it) whose dual contains $E^*$ as an $L$-summand. We also study surjective isometries on extremely non-complex Banach spaces and construct an example of a real Banach space whose group of surjective isometries reduces to $\pm Id$, but the group of surjective isometries of its dual contains the group of isometries of a separable infinite-dimensional Hilbert space as a subgroup.
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 $X$ 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…
Let $Γ$ be an infinite set equipped with the discrete topology. We prove that the space $\ell_{\infty}(Γ),$ of all complex-valued bounded functions on $Γ$, satisfies the Mazur-Ulam property, that is, every surjective isometry from the unit sphere of $\ell_{\infty}(Γ)$ onto the unit sphere of an arbitrary complex Banach space $X$ admits a unique extension to a surjective real linear isometry from $\ell_{\infty}(Γ)$ to $X$.
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 …
We show that a continuous additive positively homogeneous map from a closed not necessarily proper cone in a Banach space onto a Banach space is an open map precisely when it is surjective. This generalization of the usual Open Mapping Theorem for Banach spaces is then combined with Michael's Selection Theorem to yield the existence of a continuous bounded positively homogeneous right inverse of such a surjective map; a strong version of the usual Open Mapping Theorem is then a special case. As another consequence, an improved version of the analogue of Andô's Theorem for an ordered Banach spa…
We introduce the $H^1$-projective property, and use it to construct a Banach space $X$ such that the natural map $J:H^2(X)\otimes H^2(X) -> H^1(X\otimes X)$ is not onto.
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.
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…
For $1<p\le \infty$, we show the existence of a Banach space which is both injectively and surjectively universal for the class of all separable Banach spaces with an equivalent $p$-asymptotically uniformly smooth norm. We prove that this class is analytic complete in the class of separable Banach spaces. These results extend previous works by Kalton, Werner and Kurka in the case $p=\infty$.
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.
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…
In this paper we deal with those Banach spaces $Z$ which satisfy the Mazur--Ulam property, namely that every surjective isometry $Δ$ from the unit sphere of $Z$ to the unit sphere of any Banach space $Y$ admits an unique extension to a surjective real-linear isometry from $Z$ to $Y$. We prove that for every countable set $Γ$ with $\vert Γ\vert \geq 2$, the Banach space $\bigoplus_{γ\in Γ}^{c_0} X_γ$ satisfies the Mazur--Ulam property, whenever the Banach space $X_γ$ is strictly convex with dim$((X_γ)_{\mathbb{R}})\geq 2$ for every $γ$. Moreover we prove that the Banach space $C_0(K,X)$ satisfi…
We study 2-local reflexivity of the set of all surjective isometries between certain function spaces. We do not assume linearity for isometries. We prove that a 2-local isometry in the group of all surjective isometries on the algebra of all continuously differentiable functions on the closed unit interval with respect to several norms is a surjective isometry. We also prove that a 2-local isometry in the group of all surjective isometries on the Banach algebra of all Lipschitz functions on the closed unit interval with the sum-norm is a surjective isometry.
Let $M$ and $N$ be unital Jordan-Banach algebras, and let $M^{-1}$ and $N^{-1}$ denote the sets of invertible elements in $M$ and $N$, respectively. Suppose that $\mathfrak{M}\subseteq M^{-1}$ and $\mathfrak{N}\subseteq N^{-1}$ are clopen subsets of $M^{-1}$ and $N^{-1}$, respectively, which are closed for powers, inverses and products of the form $U_{a} (b)$. In this paper we prove that for each surjective isometry $Δ: \mathfrak{M}\to \mathfrak{N}$ there exists a surjective real-linear isometry $T_0: M\to N$ and an element $u_0$ in the McCrimmon radical of $N$ such that $Δ(a) = T_0(a) +u_0$ f…
Let $\mathbf{F}$ be a Banach space of continuous functions over a connected locally compact space $X$. We present several sufficient conditions on $\mathbf{F}$ guaranteeing that the only multiplication operators on $\mathbf{F}$ that are surjective isometries are scalar multiples of the identity. The conditions are given via the properties of the inclusion operator from $\mathbf{F}$ into $\mathcal{C}\left(X\right)$, as well as in terms of geometry of $\mathbf{F}$. An important tool in our investigation is the notion of Birkhoff Orthogonality.
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.
We prove that a surjective isometry between the unit spheres of two uniform algebras is extended to a surjective real-linear isometry between the uniform algebras. It provides the first positive solution for Tingley's problem on a Banach space of analytic functions.
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…
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 ...
Let $A$ and $B$ be unital semisimple commutative Banach algebras and $T$ a map from the invertible group $A^{-1}$ onto $B^{-1}$. Linearity and multiplicativity of the map are not assumed. We consider the hypotheses on $T$: (1) $σ(TfTg)=σ(fg)$; (2) $σ_π(TfTg-α)\cap σ_π(fg-α)\ne \emptyset$; (3) $\mathrm{r} (TfTg-α)=\mathrm{r}(fg-α)$ hold for some non-zero complex number $α$ and for every $f, g\in A^{-1}$, where $σ(\cdot)$ (resp. $σ_π(\cdot)$) denotes the (resp. peripheral) spectrum and $\rr(\cdot)$ denotes the spectral radius. Under each of the hypotheses we show representations for $T$ and unde…
In this paper, we prove that the existence of an $\varepsilon$-isometry from a separable Banach space $X$ into $Y$ (the James space or a reflexive space) implies the existence of a linear isometry from $X$ into $Y$. Then we present a set valued mapping version lemma on non-surjective $\varepsilon$-isometries of Banach spaces. Using the above results, we also discuss the rotundity and smoothness of Banach spaces under the perturbation by $\varepsilon$-isometries.
For $1<p\leqslant \infty$, we study the complexity and the existence of universal spaces for two classes of separable Banach spaces, denoted $\textsf{A}_p$ and $\textsf{N}_p$, and related to asymptotic smoothness in Banach spaces. We show that each of these classes is Borel in the class of separable Banach spaces. Then we build small families of Banach spaces that are both injectively and surjectively universal for these classes. Finally, we prove the optimality of this universality result, by proving in particular that none of these classes admits a universal space.
We prove that every JBW$^*$-triple $M$ with rank one or rank bigger than or equal to three satisfies the Mazur--Ulam property, that is, every surjective isometry from the unit sphere of $M$ onto the unit sphere of another Banach space $Y$ extends to a surjective real linear isometry from $M$ onto $Y$. We also show that the same conclusion holds if $M$ is not a JBW$^*$-triple factor, or more generally, if the atomic part of $M^{**}$ is not a rank two Cartan factor.
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.
We show in this paper that every bijective linear isometry between the continuous section spaces of two non-square Banach bundles gives rise to a Banach bundle isomorphism. This is to support our expectation that the geometric structure of the continuous section space of a Banach bundle determines completely its bundle structures. We also describe the structure of an \emph{into} isometry from a continuous section space into an other. However, we demonstrate by an example that a non-surjective linear isometry can be far away from a subbundle embedding.
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.
We introduce the Jordan-strict topology on the multipliers algebra of a JB$^*$-algebra, a notion which was missing despite the fourty years passed after the first studies on Jordan multipliers. In case that a C$^*$-algebra $A$ is regarded as a JB$^*$-algebra, the J-strict topology of $M(A)$ is precisely the well-studied C$^*$-strict topology. We prove that every JB$^*$-algebra $\mathfrak{A}$ is J-strict dense in its multipliers algebra $M(\mathfrak{A})$, and that latter algebra is J-strict complete. We show that continuous surjective Jordan homomorphisms, triple homomorphisms, and orthogonalit…
Commutativity preserving mappings in Banach algebras
Faculty of Mathematics and Physics, University of Ljubljana（卢布尔雅那大学数学物理学院）；Faculty of Natural Sciences and Mathematics, University of Maribor（马里博尔大学自然科学与数学学院）；IMFM, Ljubljana（斯洛文尼亚科学院数学物理研究所）；Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada（格拉纳达大学理学院数学分析系）
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 $$f(\frac{x+y+z}{3})+f(\frac{x-2y+z}{3})+f(\frac{x+y-2z}{3})= f(x).$$
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.
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.
