Let $f$ be a continuous map of the closure $\overline{\mathrm{\Delta }}$ of the open unit disc $\mathrm{\Delta }$ of $\mathbb{C}$ into a unital associative Banach algebra $\mathcal{A}$, whose restriction to $\mathrm{\Delta }$ is holomorphic, and which satisfies the condition whereby $0\notin \sigma \left(f\left(z\right)\right)\subset \overline{\mathrm{\Delta }}$ for all $z\in \mathrm{\Delta }$ and $\sigma \left(f\left(z\right)\right)\subset \partial \mathrm{\Delta }$ whenever $z\in \partial \mathrm{\Delta }$ (where $\sigma \left(x\right)$ is the spectrum of any $x\in \mathcal{A}$). One of the basic results of the present paper is that $f$ is , that is to say, $\sigma \left(f\left(z\right)\right)$ is then a compact subset of $\partial \mathrm{\Delta }$ that does not depend on $z$ for all $z\in \overline{\mathrm{\Delta }}$. This fact will be applied to holomorphic self-maps of the open unit ball of some $J^{*}$-algebra...

Valov proved a general version of Arvanitakis's simultaneous selection theorem which is a common generalization of both Michael's selection theorem and Dugundji's extension theorem. We show that Valov's theorem can be extended by applying an argument by means of Pettis integrals due to Repovš, Semenov and Shchepin.

An example of a nonzero σ-finite Borel measure μ with everywhere dense linear manifold ${}_{\mu }$ of admissible (in the sense of invariance) translation vectors is constructed in the Hilbert space ℓ₂ such that μ and any shift ${\mu }^{\left(a\right)}$ of μ by a vector $a\in \ell ₂{\setminus }_{\mu }$ are neither equivalent nor orthogonal. This extends a result established in [7].

A class of convex functions where the sets of subdifferentials behave like the unit ball of the dual space of an Asplund space is found. These functions, which we called Asplund functions also possess some stability properties. We also give a sufficient condition for a function to be an Asplund function in terms of the upper-semicontinuity of the subdifferential map.

Let $𝒜$ be a $\sigma$-algebra on a set $X$. If $A$ belongs to $𝒜$ let $e\left(A\right)$ be the characteristic function of $A$. Let ${\ell }_0^{\infty }(X,𝒜$ be the linear space generated by $\left\{e\right(A):A\in 𝒜\}$ endowed with the topology of the uniform convergence. It is proved in this paper that if $\left(E_n\right)$ is an increasing sequence of subspaces of ${\ell }_0^{\infty }(X,𝒜)$ covering it, there is a positive integer $p$ such that $E_p$ is a dense barrelled subspace of ${\ell }_0^{\infty }(X,𝒜)$, and some new results in measure theory are deduced from this fact.

R. Deville and J. Rodríguez proved that, for every Hilbert generated space $X$, every Pettis integrable function $f:[0,1]\to X$ is McShane integrable. R. Avilés, G. Plebanek, and J. Rodríguez constructed a weakly compactly generated Banach space $X$ and a scalarly null (hence Pettis integrable) function from $[0,1]$ into $X$, which was not McShane integrable. We study here the mechanism behind the McShane integrability of scalarly negligible functions from $[0,1]$ (mostly) into $C\left(K\right)$ spaces. We focus in more detail on the behavior...

It is proved that no convex and Fréchet differentiable function on c0(w1), whose derivative is locally uniformly continuous, attains its minimum at a unique point.