A characterization of topological spaces admitting a countable cover by sets of small local diameter close in spirit to known characterizations of fragmentability is obtained. It is proved that if X and Y are Hausdorff compacta such that C(X) has an equivalent p-Kadec norm and $C_p\left(Y\right)$ has a countable cover by sets of small local norm diameter, then $C_p\left(X\times Y\right)$ has a countable cover by sets of small local norm diameter as well.

Let $D^{\text{'}}\subset {ℂ}^{n-1}$ be a bounded domain of Lyapunov and $f(z^{\text{'}},z_n)$ a holomorphic function in the cylinder $D=D^{\text{'}}\times U_n$ and continuous on $\overline{D}$. If for each fixed $a^{\text{'}}$ in some set $E\subset \partial D^{\text{'}}$, with positive Lebesgue measure $\text{mes}E>0$, the function $f(a^{\text{'}},z_n)$ of $z_n$ can be continued to a function holomorphic on the whole plane with the exception of some finite number (polar set) of singularities, then $f(z^{\text{'}},z_n)$ can be holomorphically continued to $(D^{\text{'}}\times ℂ)\setminus S$, where $S$ is some analytic (closed pluripolar) subset of $D^{\text{'}}\times ℂ$.

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...

In this paper the hyponormal operators on Krein spaces are introduced. We state conditions for the hyponormality of bounded operators focusing, in particular, on those operators $T$ for which there exists a fundamental decomposition $𝕂={𝕂}^{+}\oplus {𝕂}^{-}$ of the Krein space $𝕂$ with ${𝕂}^{+}$ and ${𝕂}^{-}$ invariant under $T$.