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### $\left(\phi ,\varphi \right)$-derivations on semiprime rings and Banach algebras

Communications in Mathematics

Let $ℛ$ be a semiprime ring with unity $e$ and $\phi$, $\varphi$ be automorphisms of $ℛ$. In this paper it is shown that if $ℛ$ satisfies $2𝒟\left({x}^{n}\right)=𝒟\left({x}^{n-1}\right)\phi \left(x\right)+\varphi \left({x}^{n-1}\right)𝒟\left(x\right)+𝒟\left(x\right)\phi \left({x}^{n-1}\right)+\varphi \left(x\right)𝒟\left({x}^{n-1}\right)$ for all $x\in ℛ$ and some fixed integer $n\ge 2$, then $𝒟$ is an ($\phi$, $\varphi$)-derivation. Moreover, this result makes it possible to prove that if $ℛ$ admits an additive mappings $𝒟,𝒢:ℛ\to ℛ$ satisfying the relations $\begin{array}{c}2𝒟\left({x}^{n}\right)=𝒟\left({x}^{n-1}\right)\phi \left(x\right)+\varphi \left({x}^{n-1}\right)𝒢\left(x\right)+𝒢\left(x\right)\phi \left({x}^{n-1}\right)+\varphi \left(x\right)𝒢\left({x}^{n-1}\right)\phantom{\rule{0.166667em}{0ex}},\\ 2𝒢\left({x}^{n}\right)=𝒢\left({x}^{n-1}\right)\phi \left(x\right)+\varphi \left({x}^{n-1}\right)𝒟\left(x\right)+𝒟\left(x\right)\phi \left({x}^{n-1}\right)+\varphi \left(x\right)𝒟\left({x}^{n-1}\right)\phantom{\rule{0.166667em}{0ex}},\end{array}$ for all $x\in ℛ$ and some fixed integer $n\ge 2$, then $𝒟$ and $𝒢$ are ($\phi$, $\varphi$)derivations under some torsion restriction. Finally, we apply these purely ring theoretic results to semi-simple Banach algebras.

### A characteristic property of the algebra $C{\left(\text{Ω}\right)}_{\beta }$.

Sibirskij Matematicheskij Zhurnal

### A characterization of $C\left(X\right)$

Commentationes Mathematicae Universitatis Carolinae

### A characterization of essential sets of function algebras

Archivum Mathematicum

In the present note, we characterize the essential set $E$ of a function algebra $A$ defined on a compact Hausdorff space $X$ in terms of local properties of functions in $A$ at the points off $E$.

### A characterization of the invertible measures

Studia Mathematica

Let G be a locally compact abelian group and M(G) its measure algebra. Two measures μ and λ are said to be equivalent if there exists an invertible measure ϖ such that ϖ*μ = λ. The main result of this note is the following: A measure μ is invertible iff |μ̂| ≥ ε on Ĝ for some ε > 0 and μ is equivalent to a measure λ of the form λ = a + θ, where a ∈ L¹(G) and θ ∈ M(G) is an idempotent measure.

### A cheaper Swiss cheese

Studia Mathematica

### A corrected correction

Fundamenta Mathematicae

### A criterion for subharmonicity of a function of the spectrum

Studia Mathematica

### A discontinuous function does not operate on the real part of a function algebra

Časopis pro pěstování matematiky

### A domination theorem for function algebras

Colloquium Mathematicae

### A function algebra without any point derivation

Časopis pro pěstování matematiky

### A generalization of the Maligranda--Orlicz Lemma.

JIPAM. Journal of Inequalities in Pure &amp; Applied Mathematics [electronic only]

### A generalized Bishop-Gončar construction

Commentationes Mathematicae Universitatis Carolinae

### A hierarchy in the family of real surjective functions

Open Mathematics

This expository paper focuses on the study of extreme surjective functions in ℝℝ. We present several different types of extreme surjectivity by providing examples and crucial properties. These examples help us to establish a hierarchy within the different classes of surjectivity we deal with. The classes presented here are: everywhere surjective functions, strongly everywhere surjective functions, κ-everywhere surjective functions, perfectly everywhere surjective functions and Jones functions. The...

### A Maximal Algebra.

Mathematica Scandinavica

### A maximal subalgebra of R(X).

Mathematica Scandinavica

### A note on a construction of J. F. Feinstein

Studia Mathematica

In [6] J. F. Feinstein constructed a compact plane set X such that R(X), the uniform closure of the algebra of rational functions with poles off X, has no non-zero, bounded point derivations but is not weakly amenable. In the same paper he gave an example of a separable uniform algebra A such that every point in the character space of A is a peak point but A is not weakly amenable. We show that it is possible to modify the construction in order to produce examples which are also regular.

### A note on Banach algebras that are not isomorphic to a group algebra.

Revista Colombiana de Matemáticas

### A note on criteria of Le Page and Hirschfeld-Żelazko for the commutativity of Banach algebras

Studia Mathematica

### A Note on Extreme and Exposed Points in Cn(I).

Mathematische Annalen

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