The main result says that nondiscrete, weakly closed, containing no nontrivial linear subspaces, additive subgroups in separable reflexive Banach spaces are homeomorphic to the complete Erdős space. Two examples of such subgroups in ${\ell }^1$ which are interesting from the Banach space theory point of view are discussed.

Properties of topologically invertible elements and the topological spectrum of elements in unital semitopological algebras are studied. It is shown that the inversion $x\mapsto x^{-1}$ is continuous in every invertive Fréchet algebra, and singly generated unital semitopological algebras have continuous characters if and only if the topological spectrum of the generator is non-empty. Several open problems are presented.

Let be an operator ideal on LCS’s. A continuous seminorm p of a LCS X is said to be - continuous if $Q{̃}_p{\in }^{inj}(X,X{̃}_p)$, where $X{̃}_p$ is the completion of the normed space $X_p=X/p^{-1}\left(0\right)$ and $Q{̃}_p$ is the canonical map. p is said to be a Groth()- seminorm if there is a continuous seminorm q of X such that p ≤ q and the canonical map $Q{̃}_{pq}:X{̃}_q\to X{̃}_p$ belongs to $(X{̃}_q,X{̃}_p)$. It is well known that when is the ideal of absolutely summing (resp. precompact, weakly compact) operators, a LCS X is a nuclear (resp. Schwartz, infra-Schwartz) space if and only if every continuous...

Le présent article est consacré à l’étude de la topologie nucléaire associée à une topologie localement convexe séparée arbitraire et ses applications. On utilise des techniques de Bornologie. On établit que tout espace ultra-bornologique, en particulier tout espace de Banach, est dual fort d’un espace nucléaire complet et on donne quelques applications de ce résultat. Nous montrons l’existence d’une large classe d’espaces nucléaires complets à bornés métrisables et à duals forts non nucléaires...

Let be a family of compact sets in a Banach algebra A such that is stable with respect to finite unions and contains all finite sets. Then the sets $U\left(K\right):=I\in Id\left(A\right):I\cap K=\varnothing$, K ∈ define a topology τ() on the space Id(A) of closed two-sided ideals of A. is called normal if $I_i\to I$ in (Id(A),τ()) and x ∈ A╲I imply $limin{f}_i\parallel x+I_i\parallel >0$. (1) If the family of finite subsets of A is normal then Id(A) is locally compact in the hull kernel topology and if moreover A is separable then Id(A) is second countable. (2) If the family of countable compact sets...

Given a von Neumann algebra M, we consider the central extension E(M) of M. We introduce the topology t c(M) on E(M) generated by a center-valued norm and prove that it coincides with the topology of local convergence in measure on E(M) if and only if M does not have direct summands of type II. We also show that t c(M) restricted to the set E(M)h of self-adjoint elements of E(M) coincides with the order topology on E(M)h if and only if M is a σ-finite type Ifin von Neumann algebra.

We enlarge the amount of embeddings of the group G of invertible transformations of [0,1] into spaces of bounded linear operators on Orlicz spaces. We show the equality of the inherited coarse topologies.

Some topologies on the space Id(A) of two-sided and closed ideals of a Banach algebra are introduced and investigated. One of the topologies, namely ${\tau }_{\infty }$, coincides with the so-called strong topology if A is a C*-algebra. We prove that for a separable Banach algebra ${\tau }_{\infty }$ coincides with a weaker topology when restricted to the space Min-Primal(A) of minimal closed primal ideals and that Min-Primal(A) is a Polish space if ${\tau }_{\infty }$ is Hausdorff; this generalizes results from [1] and [5]. All subspaces of Id(A)...

On définit sur un espace vectoriel $E$ une classe de topologies qui rendent la multiplication continue, mais ne sont pas vectorielles en général. Sur un espace complexe $E$ elles permettent d’obtenir encore les principales propriétés des fonctions plurisousharmoniques. De telles topologies séparées sont localement pseudo-convexes (mais non localement convexes en général) : cette notion intervient dans les extensions données récemment par l’auteur du théorème de Banach-Steinhaus aux familles de polynômes...