We study generalized derivations G defined on a complex Banach algebra A such that the spectrum σ(Gx) is finite for all x ∈ A. In particular, we show that if A is unital and semisimple, then G is inner and implemented by elements of the socle of A.

It is shown that every almost linear Pexider mappings $f$, $g$, $h$ from a unital $C^{*}$-algebra $𝒜$ into a unital $C^{*}$-algebra $ℬ$ are homomorphisms when $f\left(2^{n}uy\right)=f\left(2^{n}u\right)f\left(y\right)$, $g\left(2^{n}uy\right)=g\left(2^{n}u\right)g\left(y\right)$ and $h\left(2^{n}uy\right)=h\left(2^{n}u\right)h\left(y\right)$ hold for all unitaries $u\in 𝒜$, all $y\in 𝒜$, and all $n\in \mathbb{Z}$, and that every almost linear continuous Pexider mappings $f$, $g$, $h$ from a unital $C^{*}$-algebra $𝒜$ of real rank zero into a unital $C^{*}$-algebra $ℬ$ are homomorphisms when $f\left(2^{n}uy\right)=f\left(2^{n}u\right)f\left(y\right)$, $g\left(2^{n}uy\right)=g\left(2^{n}u\right)g\left(y\right)$ and $h\left(2^{n}uy\right)=h\left(2^{n}u\right)h\left(y\right)$ hold for all $u\in \{v\in 𝒜\mid v=v^{*}\text{and}v\text{is}\text{invertible}\}$, all $y\in 𝒜$ and all $n\in \mathbb{Z}$. Furthermore, we prove the Cauchy-Rassias stability of $*$-homomorphisms between unital $C^{*}$-algebras, and $ℂ$-linear...

We study similarity to partial isometries in C*-algebras as well as their relationship with generalized inverses. Most of the results extend some recent results regarding partial isometries on Hilbert spaces. Moreover, we describe partial isometries by means of interpolation polynomials.

In a recent paper by H. X. Cao, J. H. Zhang and Z. B. Xu an $\alpha$-Lipschitz operator from a compact metric space into a Banach space $A$ is defined and characterized in a natural way in the sence that $F:K\to A$ is a $\alpha$-Lipschitz operator if and only if for each $\sigma \in X^{*}$ the mapping $\sigma \circ F$ is a $\alpha$-Lipschitz function. The Lipschitz operators algebras $L^{\alpha }(K,A)$ and $l^{\alpha }(K,A)$ are developed here further, and we study their amenability and weak amenability of these algebras. Moreover, we prove an interesting result that $L^{\alpha }(K,A)$ and $l^{\alpha }(K,A)$ are isometrically...

We describe the topological reflexive closure of the isometry group of the suspension of B(H).

We study the representation of orthogonally additive mappings acting on Hilbert C*-modules and Hilbert H*-modules. One of our main results shows that every continuous orthogonally additive mapping f from a Hilbert module W over 𝓚(𝓗) or 𝓗𝓢(𝓗) to a complex normed space is of the form f(x) = T(x) + Φ(⟨x,x⟩) for all x ∈ W, where T is a continuous additive mapping, and Φ is a continuous linear mapping.