Let A and B be unital, semisimple commutative Banach algebras with the maximal ideal spaces M A and M B, respectively, and let r(a) be the spectral radius of a. We show that if T: A → B is a surjective mapping, not assumed to be linear, satisfying r(T(a) + T(b)) = r(a + b) for all a; b ∈ A, then there exist a homeomorphism φ: M B → M A and a closed and open subset K of M B such that $$\widehat{T\left(a\right)}\left(y\right)=\left\{\begin{array}{c}\widehat{T\left(e\right)}\left(y\right)\widehat{a}\left(\phi \left(y\right)\right)y\in K\\ \widehat{T\left(e\right)}\left(y\right)\overline{\widehat{a}\left(\phi \left(y\right)\right)}y\in M_{ℬ}\setminus K\end{array}\right.$$ for all a ∈ A, where e is unit element of A. If, in addition, $$\widehat{T\left(e\right)}=1$$ and $$\widehat{T\left(ie\right)}=i$$ on M B, then T is an algebra isomorphism.

2000 Mathematics Subject Classification: 54H05, 03E15, 46B26We answer positively a question raised by S. Argyros: Given any coanalytic, nonalytic subset Σ′ of the irrationals, we construct, in Mercourakis space c1(Σ′), an adequate compact which is Gul’ko and not Talagrand. Further, given any Borel, non Fσ subset Σ′ of the irrationals, we construct, in c1(Σ′), an adequate compact which is Talagrand and not Eberlein.Supported by grants AV CR 101-90-03, and GA CR 201-01-1198

Étant donnés un compact $K$ du plan complexe, et une mesure non nulle sur $K$, on étudie $H^{\infty }\left(\mu \right)$, l’adhérence dans $L^{\infty }\left(\mu \right)$, pour la topologie $\sigma \left(L^{\infty }\right(\mu ),L^1(\mu \left)\right)$, de l’algèbre des fractions rationnelles d’une variable complexe, à pôles hors de $K$. Le résultat principal obtenu est qu’il existe un sous-ensemble $E_{\mu }$ de $K$, éventuellement vide, mesurable pour la mesure de Lebesgue plane, et une mesure ${\mu }_s$, éventuellement nulle, absolument continue par rapport à la mesure $\mu$, tels que : $H^{\infty }\left(\mu \right)$ soit isométriquement isomorphe à $H^{\infty }({\lambda }_{E_{\mu }})\oplus L^{\infty }({\mu }_s)$, où ${\lambda }_{E_{\mu }}$ désigne la...

With the help of recent adjacent dyadic constructions by Hytönen and the author, we give an alternative proof of results of Lechner, Müller and Passenbrunner about the $L^p$-boundedness of shift operators acting on functions $f\in L^p(X;E)$ where 1 < p < ∞, X is a metric space and E is a UMD space.

For a given bi-continuous semigroup ${\left(T\left(t\right)\right)}_{t\ge 0}$ on a Banach space $X$ we define its adjoint on an appropriate closed subspace $X^{\circ }$ of the norm dual $X^{\text{'}}$. Under some abstract conditions this adjoint semigroup is again bi-continuous with respect to the weak topology $\sigma (X^{\circ },X)$. We give the following application: For $\Omega$ a Polish space we consider operator semigroups on the space ${\mathrm{C}}_{\mathrm{b}}\left(\Omega \right)$ of bounded, continuous functions (endowed with the compact-open topology) and on the space $\mathrm{M}\left(\Omega \right)$ of bounded Baire measures (endowed with the weak${}^{*}$-topology)....

We construct a metrizable simplex X such that for each n ɛ ℕ there exists a bounded function f on ext X of Baire class n that cannot be extended to a strongly affine function of Baire class n. We show that such an example cannot be constructed via the space of harmonic functions.

Let 𝒳 be a compact Hausdorff space which satisfies the first axiom of countability, I = [0,1] and 𝓒(𝒳,I) the set of all continuous functions from 𝒳 to I. If φ: 𝓒(𝒳,I) → 𝓒(𝒳,I) is a bijective affine map then there exists a homeomorphism μ: 𝒳 → 𝒳 such that for every component C in 𝒳 we have either φ(f)(x) = f(μ(x)), f ∈ 𝓒(𝒳,I), x ∈ C, or φ(f)(x) = 1-f(μ(x)), f ∈ 𝓒(𝒳,I), x ∈ C.

For every n ≥ 2, let cc(ℝⁿ) denote the hyperspace of all nonempty compact convex subsets of the Euclidean space ℝⁿ endowed with the Hausdorff metric topology. Let cb(ℝⁿ) be the subset of cc(ℝⁿ) consisting of all compact convex bodies. In this paper we discover several fundamental properties of the natural action of the affine group Aff(n) on cb(ℝⁿ). We prove that the space E(n) of all n-dimensional ellipsoids is an Aff(n)-equivariant retract of cb(ℝⁿ). This is applied to show that cb(ℝⁿ) is homeomorphic...