Narrow Results

Let $𝔘$ be a $\varphi$-Johnson amenable Banach algebra in which $\varphi$ is a nonzero multiplicative linear functional on $𝔘$. Suppose that $X$ is a Banach $𝔘$-bimodule such that $a.x=\varphi (a)x$ for all $a\in 𝔘$, $x\in X$ or $x.a=\varphi (a)x$ for all $a\in 𝔘$, $x\in X$. We show that every continuous Jordan derivation from $𝔘$ to $X$ is a derivation, and every continuous Lie derivation from $𝔘$ to $X$ decomposed into the sum of a continuous derivation and a continuous center-valued trace. Then we apply our results for character amenable Banach algebras and amenable Banach algebras. We also provide some results about $\varphi$-Johnson amenability, especially we give some conditions equivalent to $\varphi$-Johnson amenability.

Let ${𝒜}$ be a Banach algebra, and let $a,b\in {𝒜}$ satisfying $a^k{b}^k{a}^k=a^{k+1}$ for some $k\in \mathbb{N}$. Under this condition, we show that Cline’s formula holds for some new generalized inverses, so $a$ has strong Drazin inverse (respectively, generalized strong Drazin inverse, Hirano inverse and generalized Hirano inverse) if and only if $b^k{a}^k$ has strong Drazin inverse (respectively, generalized strong Drazin inverse, Hirano inverse and generalized Hirano inverse). As a particular case, some results for Jacobson’s lemma are given.

In this paper, with the help of a variant of Schauder fixed point theorem in the real Banach algebra together with the finite difference method (FDM), we take a brief look at the $p$-adic analog of Richards’ equation derived by Khrennikov et al. [Application of $p$-adic wavelets to model reaction–diffusion dynamics in random porous media, J. Fourier Anal. Appl.22 (2016) 809–822], and study the solvability and solution of this problem. This equation is formulated by a kinetic equation during the modeling of the reaction–diffusion dynamics in random porous media. Moreover, in order to guarantee the convergence of the presented iterative schemes, some sufficient conditions would be presented.

Let $R$ be an associative ring with center $Z(R).$ The objective of this paper is to discuss the commutativity of a semiprime ring $R$ which admits a derivation $d$ such that $d([x^m,y^n])\pm [x^m,y^n]\in Z(R)$ for all $x,y\in R$ or $d([x^m,y^n])\in Z(R)$ for all $x,y\in R$ or $d(x^m\circ y^n)\in Z(R)$ for all $x,y\in R,$ where $m$ and $n$ are fixed positive integers. Finally, we apply these purely ring theoretic results to obtain commutativity of Banach algebra via derivation.

Let ${𝒜}$ and $𝔄$ be two Banach algebras such that ${𝒜}$ is a Banach $𝔄$-bimodule with the left and right compatible action of $𝔄$ on ${𝒜}$. Let ${𝒜}⋊𝔄$ be a strongly splitting Banach algebra extension of $𝔄$ by ${𝒜}$. We show that (super) amenability of ${𝒜}⋊𝔄$ implies (super) module amenability of ${𝒜}$ and (super) amenability $𝔄$. We investigate biprojectivity and biflatness of ${𝒜}⋊𝔄$ in the some especial cases. We also give some results related to module biprojectivity and module biflatness of ${𝒜}$, when ${𝒜}⋊𝔄$ is biprojective or biflat.

Let ${𝒜}$ and $𝔄$ be two $C^{\ast }$-algebras such that ${𝒜}$ is a Banach $𝔄$-bimodule with the left and right compatible action of $𝔄$ on ${𝒜}$. We define ${𝒜}⋊𝔄$ as a $C^{\ast }$-algebra, where it is a strongly splitting $C^{\ast }$-algebra extension of $𝔄$ by ${𝒜}$. Normal, self-adjoint, unitary, invertible and projection elements of ${𝒜}⋊𝔄$ are characterized; sufficient and necessary conditions for existing unit and bounded approximate identity of ${𝒜}⋊𝔄$ as a Banach algebra and as a $C^{\ast }$-algebra are given. We characterize ∗-automorphisms on ${𝒜}⋊𝔄$ and give some results related to ∗-homomorphisms, ∗-representations and completely bounded maps on this $C^{\ast }$-algebra. Also, we have constructed a new Hilbert $C^{\ast }$-module $X⋊Y$ over ${𝒜}⋊𝔄$, where $X$ is a Hilbert $C^{\ast }$-module over ${𝒜}$ and $Y$ is a Hilbert $C^{\ast }$-module over $𝔄$.

In this paper, we introduce and investigate three new versions of the Rickart condition for rings. These conditions, as well as, three new corresponding regularities are defined using projection invariance. We show how these conditions relate to each other as well as their connections to the well-known Baer, Rickart, quasi-Baer, p.q.-Baer, regular, and biregular conditions. Applications to polynomial extensions and to triangular and full matrix rings are provided. Examples illustrate and delimit results.

In this paper, we study and compare the types of diagonals for Lau product Banach algebras ${𝒜}{\times }_{\phi }{ℬ}$ and diagonals of two Banach algebras ${𝒜}$ and ${ℬ}$, where $\phi$ is a homomorphism from ${ℬ}$ to ${𝒜}$. As a consequence, we obtain the (approximate) amenability and contractibility ${𝒜}{\times }_{\phi }{ℬ}$ in terms of (approximate) amenability and contractibility of ${𝒜}$ and ${ℬ}$. Moreover, we introduce the notion of pseudo-virtual diagonal for a Banach algebra and present some characterization results for Lau product Banach algebras.

In this paper, we define the Jordan notions of contractibility and biprojectivity of Banach algebras and comparing to the classical notions of contractibility and biprojectivity. Also, we introduce the notion of Jordan diagonal and find relations between these Jordan notions. As a main result, we show that the Banach algebra $𝔄$ is unital and Jordan contractible if and only if it has a Jordan diagonal if and only if $𝔄$ is unital and Jordan biprojective, whenever $𝔄\widehat{\otimes }𝔄$ is a Banach $(𝔄,{𝔄}^{\mathrm{\text{op}}})$-bimodule.

Let $\mathrm{\Lambda }$ be a direct set, ${\{A_{\alpha }\}}_{\alpha \in \mathrm{\Lambda }}$ be a family of Banach algebras with bounded approximate identity (or unital) and $I$ be a set. We consider the Banach algebra ${\ell }^{\infty }({\ell }_{A_{\beta }}(I,A_{\alpha }))$. We show that this algebra has a bounded approximate identity (or is unital) if and only if $I$ is finite. We also characterize the left multipliers of these algebras and investigate their amenability of them. Moreover, we characterize the character spaces (Gelfand spaces) of these algebras in a special case.

In this paper, we prove the generalized Hyers–Ulam stability of C*-ternary quadratic higher derivations of any rank by using the Banach fixed point theorem.

Let R be a prime ring, U the Utumi quotient ring of R, C = Z(U) the extended centroid of R, L a non-central Lie ideal of R, and H, G nonzero generalized derivations of R. Suppose that there exists an integer n ≥ 1 such that H(uⁿ)uⁿ + uⁿG(uⁿ) ∈ C for all u ∈ L, then either there exists a ∈ U such that H(x) = xa and G(x) = -ax, or R satisfies the standard identity s₄ and one of the following holds: (i) char(R) = 2; (ii) n is even and there exist a′ ∈ U, α ∈ C and derivations d, δ of R such that H(x) = a′ x + d(x) and G(x) = (α-a′)x + δ(x); (iii) n is even and there exist a′ ∈ U and a derivation δ of R such that H(x)=xa′ and G(x) = -a′ x + δ(x); (iv) n is odd and there exist a′, b′ ∈ U and α, β ∈ C such that H(x) = a′ x + x(β-b′) and G(x) = b′ x+x(α-a′); (v) n is odd and there exist α, β ∈ C and a derivation d of R such that H(x) = α x+d(x) and G(x) = β x + d(x); (vi) n is odd and there exist a′ ∈ U and α ∈ C such that H(x) = xa′ and G(x) = (α - a′)x. As an application of this purely algebraic result, we obtain some range inclusion results of continuous or spectrally bounded generalized derivations H and G on Banach algebras R satisfying the condition H(xⁿ)xⁿ + xⁿG(xⁿ) ∈ rad(R) for all x ∈ R, where rad(R) is the Jacobson radical of R.

For a tuple A = (A₁, A₂, …, A_n) of elements in a unital algebra over ℂ, its projective spectrumP(A) or p(A) is the collection of z ∈ ℂⁿ, or respectively z ∈ ℙ^n-1 such that A(z) = z₁A₁ + z₂A₂ + ⋯ + z_nA_n is not invertible in . In finite dimensional case, projective spectrum is a projective hypersurface. When A is commuting, P(A) looks like a bundle over the Taylor spectrum of A. In the case is reflexive or is a C*-algebra, the projective resolvent setP^c(A) := ℂⁿ \ P(A) is shown to be a disjoint union of domains of holomorphy. -valued 1-form A^-1(z)dA(z) reveals the topology of P^c(A), and a Chern–Weil type homomorphism from invariant multilinear functionals to the de Rham cohomology is established.

It is shown that the connected stable rank of a unital Banach _*-algebra containing two isometries with orthogonal ranges is infinity. Several consequences as its applications and variations are also obtained.

In this paper we prove, among other things, that a semi-simple and topologically simple Banach algebra which has an identity and contains a non-zero finite-dimensional element is itself a finite-dimensional algebra.

The existence of support sets in a real topological vector spaces depends upon the existance of certain types of linear continuous functionals, which are called support functionals. In this paper we will show some applications of the support functionals in connection with approximation on C*–algebra.