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Several papers have, as their raison d'etre, the exploration of the generalised Lau product associated to a homomorphism T : B → A of Banach algebras. In this short note, we demonstrate that the generalised Lau product is isomorphic as a Banach algebra to the usual direct product A ⊕ B. We also correct some misleading claims made about the relationship between this generalised Lau product and an older construction of Monfared ['On certain products of Banach algebras with applications to harmonic analysis', Studia Math. 178(3) (2007), 277-294]. [ABSTRACT FROM AUTHOR]

In this paper we provide some bounds for the quantity $\Vert f(y)-f(x)\Vert$ , where $f:D\rightarrow \mathbb{C}$ is an analytic function on the domain $D\subset \mathbb{C}$ and $x$ , $y\in {\mathcal{B}}$ , a Banach algebra, with the spectra $\unicode[STIX]{x1D70E}(x)$ , $\unicode[STIX]{x1D70E}(y)\subset D$. Applications for the exponential and logarithmic functions on the Banach algebra ${\mathcal{B}}$ are also given. [ABSTRACT FROM AUTHOR]

In this paper we prove the following result: let $m,n\geq 1$ be distinct integers, let $R$ be an $mn(m+n)|m-n|$-torsion free semiprime ring and let $D:R\rightarrow R$ be an $(m,n)$-Jordan derivation, that is an additive mapping satisfying the relation $(m+n)D(x^{2})=2mD(x)x+2nxD(x)$ for $x\in R$. Then $D$ is a derivation which maps $R$ into its centre. [ABSTRACT FROM PUBLISHER]

In this paper, for an arbitrary $\ell ^{1}$-Munn algebra $\mathfrak{A}$ over a Banach algebra $A$ with a sandwich matrix $P$, we characterise all homomorphisms from $\mathfrak{A}$ to a commutative Banach algebra $B$. Especially, we study the character space of this algebra. Then, as an application, its character amenability is investigated. Finally, we apply these results to certain semigroups, which are called Rees matrix semigroups. [ABSTRACT FROM PUBLISHER]

The aim of this paper is to discuss the commutativity of a Banach algebra $A$ via its derivations. In particular, we prove that if $A$ is a unital prime Banach algebra and $A$ has a nonzero continuous linear derivation $d:A\rightarrow A$ such that either $d((xy)^{m})-x^{m}y^{m}$ or $d((xy)^{m})-y^{m}x^{m}$ is in the centre of $A$ for an integer $m=m(x,y)$ and sufficiently many $x,y$, then $A$ is commutative. We give examples to illustrate the scope of the main results and show that the hypotheses are not superfluous. [ABSTRACT FROM AUTHOR]

We prove that a Banach algebra B that is a completion of the universal enveloping algebra of a finite-dimensional complex Lie algebra $\mathfrak {g}$ satisfies a polynomial identity if and only if the nilpotent radical $\mathfrak {n}$ of $\mathfrak {g}$ is associatively nilpotent in B. Furthermore, this holds if and only if a certain polynomial growth condition is satisfied on $\mathfrak {n}$. [ABSTRACT FROM AUTHOR]

We prove that for a Banach algebra A having a bounded $\mathcal {Z}(A)$ -approximate identity and for every $\mathbf {[IN]}$ group G with a weight w which is either constant on conjugacy classes or satisfies $w \geq 1$ , $\mathcal {Z}(L^{1}_{w}(G) \otimes ^{\gamma } A) \cong \mathcal {Z}(L^{1}_{w}(G)) \otimes ^{\gamma } \mathcal {Z}(A)$. As an application, we discuss the conditions under which $\mathcal {Z}(L^{1}_{\omega }(G,A))$ enjoys certain Banach algebraic properties, such as weak amenability or semisimplicity. [ABSTRACT FROM AUTHOR]

Let A be a Banach algebra and let X be a Banach A-bimodule. We consider the Banach algebra ${A\oplus _1 X}$ , where A is a commutative Banach algebra. We investigate the Bochner–Schoenberg–Eberlein (BSE) property and the BSE module property on $A\oplus _1 X$. We show that the module extension Banach algebra $A\oplus _1 X$ is a BSE Banach algebra if and only if A is a BSE Banach algebra and $X=\{0\}$. Furthermore, we consider $A\oplus _1 X$ as a Banach $A\oplus _1 X$ -module and characterise the BSE module property on $A\oplus _1 X$. We show that $A\oplus _1 X$ is a BSE Banach $A\oplus _1 X$ -module if and only if A and X are BSE Banach A-modules. [ABSTRACT FROM AUTHOR]

We establish inequalities of Jensen's and Slater's type in the general setting of a Hermitian unital Banach $\ast$ -algebra, analytic convex functions and positive normalised linear functionals. [ABSTRACT FROM AUTHOR]

We study the second dual algebra of a Banach algebra and related problems. We resolve some questions raised by Ülger, which are related to Arens products. We then discuss a question of Gulick on the radical of the second dual algebra of the group algebra of a discrete abelian group and give an application of Arens regularity to Fourier and Fourier–Stieltjes transforms. [ABSTRACT FROM AUTHOR]

Let $S$ be a discrete inverse semigroup, $l^{1}(S)$ the Banach semigroup algebra on $S$ and $\mathbb{X}$ a Banach $l^{1}(S)$ -bimodule which is an $L$ -embedded Banach space. We show that under some mild conditions ${\mathcal{H}}^{1}(l^{1}(S),\mathbb{X})=0$. We also provide an application of the main result. [ABSTRACT FROM AUTHOR]

We show that an essentially amenable Banach algebra need not have an approximate identity. This answers a question posed by Ghahramani and Loy ['Generalized notions of amenability', J. Funct. Anal. 208 (2004), 229–260]. Essentially Connes-amenable dual Banach algebras are introduced and studied. [ABSTRACT FROM AUTHOR]

Extending recent results by Cascales et al. [‘Plasticity of the unit ball of a strictly convex Banach space’, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat.110(2) (2016), 723–727], we demonstrate that for every Banach space $X$ and every collection $Z_{i},i\in I$, of strictly convex Banach spaces, every nonexpansive bijection from the unit ball of $X$ to the unit ball of the sum of $Z_{i}$ by $\ell _{1}$ is an isometry. [ABSTRACT FROM AUTHOR]

We focus on a question raised by Daws ['Arens regularity of the algebra of operators on a Banach space', Bull. Lond. Math. Soc. 36 (2004), 493-503] concerning the Arens regularity of B(X), the algebra of operators on a Banach space X. Among other things, we show that B(X) is Arens regular if and only if X is ultrareflexive. [ABSTRACT FROM AUTHOR]

We show that, under special hypotheses, each 3-Jordan homomorphism ${\it\varphi}$ between Banach algebras ${\mathcal{A}}$ and ${\mathcal{B}}$ is a 3-homomorphism. [ABSTRACT FROM PUBLISHER]

Let ${\it\varphi}$ be a homomorphism from a Banach algebra ${\mathcal{B}}$ to a Banach algebra ${\mathcal{A}}$. We define a multiplication on the Cartesian product space ${\mathcal{A}}\times {\mathcal{B}}$ and obtain a new Banach algebra ${\mathcal{A}}\times _{{\it\varphi}}{\mathcal{B}}$. We show that biprojectivity as well as biflatness of ${\mathcal{A}}\times _{{\it\varphi}}{\mathcal{B}}$ are stable with respect to ${\it\varphi}$. [ABSTRACT FROM PUBLISHER]