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Let be the class of Banach spaces X for which every weakly quasi-continuous mapping f: A → X defined on an α-favorable space A is norm continuous at the points of a dense $G_{\delta }$ subset of A. We will show that this class is stable under c₀-sums and ${\ell }^p$-sums of Banach spaces for 1 ≤ p < ∞.

Let $X$ be a Baire space, $Y$ be a compact Hausdorff space and $\varphi :X\to C_p\left(Y\right)$ be a quasi-continuous mapping. For a proximal subset $H$ of $Y\times Y$ we will use topological games ${𝒢}_1\left(H\right)$ and ${𝒢}_2\left(H\right)$ on $Y\times Y$ between two players to prove that if the first player has a winning strategy in these games, then $\varphi$ is norm continuous on a dense $G_{\delta }$ subset of $X$. It follows that if $Y$ is Valdivia compact, each quasi-continuous mapping from a Baire space $X$ to $C_p\left(Y\right)$ is norm continuous on a dense $G_{\delta }$ subset of $X$.

We consider the question of preservation of Baire and weakly Baire category under images and preimages of certain kind of functions. It is known that Baire category is preserved under image of quasi-continuous feebly open surjections. In order to extend this result, we introduce a strictly larger class of quasi-continuous functions, i.e. the class of quasi-interior continuous functions. We show that Baire and weakly Baire categories are preserved under image of feebly open quasi-interior continuous...

We present a new method for studying the numerical radius of bounded operators on Hilbert $C^{*}$-modules. Our method enables us to obtain some new results and generalize some known theorems for bounded operators on Hilbert spaces to bounded adjointable operators on Hilbert $C^{*}$-module spaces.