Scattered elements of Banach algebras

Peng Cao

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Displaying similar documents to “Scattered elements of Banach algebras”

Derivations with a hereditary domain, II

A. Villena (1998)

Studia Mathematica

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The nilpotency of the separating subspace of an everywhere defined derivation on a Banach algebra is an intriguing question which remains still unsolved, even for commutative Banach algebras. On the other hand, closability of partially defined derivations on Banach algebras is a fundamental problem motivated by the study of time evolution of quantum systems. We show that the separating subspace S(D) of a Jordan derivation defined on a subalgebra B of a complex Banach algebra A satisfies...

Rings in which elements are sum of a central element and an element in the Jacobson radical

Guanglin Ma, Yao Wang, André Leroy (2024)

Czechoslovak Mathematical Journal

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An element in a ring $R$ is called CJ if it is of the form $c + j$ , where $c$ belongs to the center and $j$ is an element from the Jacobson radical. A ring $R$ is called CJ if each element of $R$ is CJ. We establish the basic properties of CJ rings, give several characterizations of these rings, and connect this notion with many standard elementwise properties such as clean, uniquely clean, nil clean, CN, and CU. We study the behavior of this notion under various ring extensions. In particular, we show...

A theorem of Gel'fand-Mazur type

Hung Le Pham (2009)

Studia Mathematica

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Denote by any set of cardinality continuum. It is proved that a Banach algebra A with the property that for every collection $a_{α} : α \in \subset A$ there exist α ≠ β ∈ such that $a_{α} \in a_{β} A^{}$ is isomorphic to $⨁_{i = 1}^{r} (ℂ [X] / X^{d_{i}} ℂ [X]) \oplus E$ , where $d ₁, . . ., d_{r} \in ℕ$ , and E is either $X ℂ [X] / X^{d ₀} ℂ [X]$ for some d₀ ∈ ℕ or a 1-dimensional $⨁_{i = 1}^{r} ℂ [X] / X^{d_{i}} ℂ [X]$ -bimodule with trivial right module action. In particular, ℂ is the unique non-zero prime Banach algebra satisfying the above condition.

Steiner forms

Jan Hora (2016)

Commentationes Mathematicae Universitatis Carolinae

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A trilinear alternating form on dimension $n$ can be defined based on a Steiner triple system of order $n$ . We prove some basic properties of these forms and using the radical polynomial we show that for dimensions up to $15$ nonisomorphic Steiner triple systems provide nonequivalent forms over $G F (2)$ . Finally, we prove that Steiner triple systems of order $n$ with different number of subsystems of order $(n - 1) / 2$ yield nonequivalent forms over $G F (2)$ .

Principal blocks and $p$ -radical groups

Xiaohan Hu, Jiwen Zeng (2016)

Czechoslovak Mathematical Journal

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Let $G$ be a finite group and $k$ a field of characteristic $p > 0$ . In this paper, we obtain several equivalent conditions to determine whether the principal block $B_{0}$ of a finite $p$ -solvable group $G$ is $p$ -radical, which means that $B_{0}$ has the property that $e_{0} {(k_{P})}^{G}$ is semisimple as a $k G$ -module, where $P$ is a Sylow $p$ -subgroup of $G$ , $k_{P}$ is the trivial $k P$ -module, ${(k_{P})}^{G}$ is the induced module, and $e_{0}$ is the block idempotent of $B_{0}$ . We also give the complete classification of a finite $p$ -solvable group $G$ which has not more...

C(X) vs. C(X) modulo its socle

F. Azarpanah, O. A. S. Karamzadeh, S. Rahmati (2008)

Colloquium Mathematicae

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Let $C_{F} (X)$ be the socle of C(X). It is shown that each prime ideal in $C (X) / C_{F} (X)$ is essential. For each h ∈ C(X), we prove that every prime ideal (resp. z-ideal) of C(X)/(h) is essential if and only if the set Z(h) of zeros of h contains no isolated points (resp. int Z(h) = ∅). It is proved that $d i m (C (X) / C_{F} (X)) \geq d i m C (X)$ , where dim C(X) denotes the Goldie dimension of C(X), and the inequality may be strict. We also give an algebraic characterization of compact spaces with at most a countable number of nonisolated points....

The kh-socle of a commutative semisimple Banach algebra

Youness Hadder (2020)

Mathematica Bohemica

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Let $𝒜$ be a commutative complex semisimple Banach algebra. Denote by $kh (soc (𝒜))$ the kernel of the hull of the socle of $𝒜$ . In this work we give some new characterizations of this ideal in terms of minimal idempotents in $𝒜$ . This allows us to show that a “result” from Riesz theory in commutative Banach algebras is not true.

Generalized derivations with power values on rings and Banach algebras

Abderrahman Hermas, Abdellah Mamouni, Lahcen Oukhtite (2024)

Mathematica Bohemica

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Let $R$ be a prime ring and $I$ a nonzero ideal of $R .$ The purpose of this paper is to classify generalized derivations of $R$ satisfying some algebraic identities with power values on $I .$ More precisely, we consider two generalized derivations $F$ and $H$ of $R$ satisfying one of the following identities: (1) ...

Piecewise hereditary algebras under field extensions

Jie Li (2021)

Czechoslovak Mathematical Journal

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Let $A$ be a finite-dimensional $k$ -algebra and $K / k$ be a finite separable field extension. We prove that $A$ is derived equivalent to a hereditary algebra if and only if so is $A \otimes_{k} K$ .

On monotonic functions from the unit interval into a Banach space with uncountable sets of points of discontinuity

Artur Michalak (2003)

Studia Mathematica

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We say that a function f from [0,1] to a Banach space X is increasing with respect to E ⊂ X* if x* ∘ f is increasing for every x* ∈ E. We show that if f: [0,1] → X is an increasing function with respect to a norming subset E of X* with uncountably many points of discontinuity and Q is a countable dense subset of [0,1], then (1) $\bar{l i n f ([0, 1])}$ contains an order isomorphic copy of D(0,1), (2) $\bar{l i n f (Q)}$ contains an isomorphic copy of C([0,1]), (3) $\bar{l i n f ([0, 1])} / \bar{l i n f (Q)}$ contains an isomorphic copy of c₀(Γ) for some uncountable...

On Some Properties of Separately Increasing Functions from [0,1]ⁿ into a Banach Space

Artur Michalak (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

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We say that a function f from [0,1] to a Banach space X is increasing with respect to E ⊂ X* if x* ∘ f is increasing for every x* ∈ E. A function $f : {[0, 1]}^{m} \to X$ is separately increasing if it is increasing in each variable separately. We show that if X is a Banach space that does not contain any isomorphic copy of c₀ or such that X* is separable, then for every separately increasing function $f : {[0, 1]}^{m} \to X$ with respect to any norming subset there exists a separately increasing function $g : {[0, 1]}^{m} \to ℝ$ such that the sets of...

Sequentially Right Banach spaces of order $p$

Mahdi Dehghani, Mohammad B. Dehghani, Mohammad S. Moshtaghioun (2020)

Commentationes Mathematicae Universitatis Carolinae

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We introduce and study two new classes of Banach spaces, the so-called sequentially Right Banach spaces of order $p$ , and those defined by the dual property, the sequentially Right $^{*}$ Banach spaces of order $p$ for $1 \leq p \leq \infty$ . These classes of Banach spaces are characterized by the notions of $L_{p}$ -limited sets in the corresponding dual space and $R_{p}^{*}$ subsets of the involved Banach space, respectively. In particular, we investigate whether the injective tensor product of a Banach space $X$ and a reflexive Banach...

C*-algebras have a quantitative version of Pełczyński's property (V)

Hana Krulišová (2017)

Czechoslovak Mathematical Journal

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A Banach space $X$ has Pełczyński’s property (V) if for every Banach space $Y$ every unconditionally converging operator $T : X \to Y$ is weakly compact. H. Pfitzner proved that $C^{*}$ -algebras have Pełczyński’s property (V). In the preprint (Krulišová, (2015)) the author explores possible quantifications of the property (V) and shows that $C (K)$ spaces for a compact Hausdorff space $K$ enjoy a quantitative version of the property (V). In this paper we generalize this result by quantifying Pfitzner’s theorem. Moreover,...

$Δ$ -weak character amenability of certain Banach algebras

Hamid Sadeghi (2019)

Archivum Mathematicum

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In this paper we investigate $Δ$ -weak character amenability of certain Banach algebras such as projective tensor product $A \hat{\otimes} B$ and Lau product $A \times_{θ} B$ , where $A$ and $B$ are two arbitrary Banach algebras and $θ \in Δ (B)$ , the character space of $B$ . We also investigate $Δ$ -weak character amenability of abstract Segal algebras and module extension Banach algebras.

On the Aronszajn property for integral equations in Banach space

Stanisław Szufla (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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For the integral equation (1) below we prove the existence on an interval $J=[0,a]$ of a solution $x$ with values in a Banach space $E$ , belonging to the class $L^{p}(J,E)$ , $p>1$ . Further, the set of solutions is shown to be a compact one in the sense of Aronszajn.