A Banach space $X$ has Pełczyński's property (V) if for every Banach space $Y$ every unconditionally converging operator $T\colon X\to Y$ 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, we prove that in dual Banach spaces a quantitative version of the property (V) implies a quantitative version of the Grothendieck property., Hana Krulišová., and Obsahuje bibliografii
It is shown that every almost linear Pexider mappings $f$, $g$, $h$ from a unital $C^*$-algebra $\mathcal A$ into a unital $C^*$-algebra $\mathcal B$ are homomorphisms when $f(2^n uy)=f(2^n u)f(y)$, $g(2^n uy)=g(2^nu)g(y)$ and $h(2^n uy)=h(2^n u)h(y)$ hold for all unitaries $u \in \mathcal A$, all $y \in \mathcal A$, and all $n\in \mathbb{Z}$, and that every almost linear continuous Pexider mappings $f$, $g$, $h$ from a unital $C^*$-algebra $\mathcal A$ of real rank zero into a unital $C^*$-algebra $\mathcal B$ are homomorphisms when $f(2^n uy)=f(2^n u)f(y)$, $g(2^n uy)=g(2^n u)g(y)$ and $h(2^n uy)=h(2^n u)h(y)$ hold for all $u \in \lbrace v\in \mathcal A\mid v=v^*\hspace{5.0pt}\text{and}\hspace{5.0pt}v\hspace{5.0pt}\text{is} \text{invertible}\rbrace $, all $y\in \mathcal A$ and all $n\in \mathbb{Z}$. Furthermore, we prove the Cauchy-Rassias stability of $*$-homomorphisms between unital $C^*$-algebras, and $\mathbb{C}$-linear $*$-derivations on unital $C^*$-algebras.