Let $G$ be a locally compact group. Let $L_t$ be the left translation in $L^{\infty }\left(G\right)$, given by $L_{t}f\left(x\right)=f\left(tx\right)$. We characterize (undre a mild set-theoretical hypothesis) the functions $f\in L^{\infty }\left(G\right)$ such that the map $t\to L_{t}f$ from $G$ into $L^{\infty }\left(G\right)$ is scalarly measurable (i.e. for $\phi \in L^{\infty }(G{)}^{*}$, $t\to \phi \left(L_{t}f\right)$ is measurable). We show that it is the case when $t\to \theta \left(L_{f}t\right)$ is measurable for each character $\theta$, and if $G$ is compact, if and only if $f$ is Riemann-measurable. We show that $t\to L_{t}f$ is Borel measurable if and only if $f$ is left uniformly continuous.Some of the measure-theoretic tools used there...

Let $A$ be a uniformly closed and locally m-convex $\Phi$-algebra. We obtain internal conditions on $A$ stated in terms of its closed ideals for $A$ to be isomorphic and homeomorphic to $C_k\left(X\right)$, the $\Phi$-algebra of all the real continuous functions on a normal topological space $X$ endowed with the compact convergence topology.

We establish interpolation properties under limiting real methods for a class of closed ideals including weakly compact operators, Banach-Saks operators, Rosenthal operators and Asplund operators. We show that they behave much better than compact operators.

Let ℬ be a Banach algebra of bounded linear operators on a Banach space X. If S is a closed operator in X such that (λ - S)^{-1} ∈ ℬ for some number λ, then S is affiliated with ℬ. The object of this paper is to study the spectral theory and Fredholm theory relative to ℬ of an operator which is affiliated with ℬ. Also, applications are given to semigroups of operators which are contained in ℬ.

Conditions involving closed range of multipliers on general Banach algebras are studied. Numerous conditions equivalent to a splitting A = TA ⊕ kerT are listed, for a multiplier T defined on the Banach algebra A. For instance, it is shown that TA ⊕ kerT = A if and only if there is a commuting operator S for which T = TST and S = STS, that this is the case if and only if such S may be taken to be a multiplier, and that these conditions are also equivalent to the existence of a factorization T = PB,...

We show that zero-dimensional nondiscrete closed subgroups do exist in Banach spaces E. This happens exactly when E contains an isomorphic copy of $c_0$. Other results on subgroups of linear spaces are obtained.

We exhibit the first examples of Fréchet spaces which contain a closed infinite dimensional subspace of universal series, but no restricted universal series. We consider classical Fréchet spaces of infinitely differentiable functions which do not admit a continuous norm. Furthermore, this leads us to establish some more general results for sequences of operators acting on Fréchet spaces with or without a continuous norm. Additionally, we give a characterization of the existence of a closed subspace...

Let $X$ be a metric space with a doubling measure, $Y$ be a boundedly compact metric space and $u:X\to Y$ be a Lebesgue precise mapping whose upper gradient $g$ belongs to the Lorentz space $L_{m,1}$, $m\ge 1$. Let $E\subset X$ be a set of measure zero. Then ${\widehat{\mathscr{H}}}_m(E\cap u^{-1}\left(y\right))=0$ for ${\mathscr{H}}_m$-a.e. $y\in Y$, where ${\mathscr{H}}_m$ is the $m$-dimensional Hausdorff measure and ${\widehat{\mathscr{H}}}_m$ is the $m$-codimensional Hausdorff measure. This property is closely related to the coarea formula and implies a version of the Eilenberg inequality. The result relies on estimates of Hausdorff content of level sets...