Search Results (5)

A topological abelian group G is said to have the quasi-convex compactness property (briefly, qcp) if the quasi-convex hull of every compact subset of G is again compact. In this paper we prove that there exist locally quasi-convex metrizable complete groups G which endowed with the weak topology associated to their character groups $G^{\wedge }$, do not have the qcp. Thus, Krein’s Theorem, a well known result in the framework of locally convex spaces, cannot be fully extended to locally quasi-convex groups. Some features of the qcp are also studied. Full article

For an abelian topological group G, the sequence group ${\ell }^1\left(G\right)$ of all absolutely summable sequences in G is studied. It is shown that ${\ell }^1\left(G\right)$ is a Pontryagin reflexive group in case G is a reflexive metrizable group or an LCA group. Further, ${\ell }^1\left(G\right)$ has the Schur property if and only if G has it and ${\ell }^1\left(G\right)$ is a Schwartz group if and only if G is linearly topologized. Full article

It is well known that every normed (even quasibarrelled) space is a Mackey space. However, in the more general realm of locally quasi-convex abelian groups an analogous result does not hold. We give the first examples of normed spaces which are not Mackey groups. Full article

For a locally quasi-convex topological abelian group (G,τ), we study the poset (mathscr{C}(G,τ)) of all locally quasi-convex topologies on (G) that are compatible with (τ) (i.e., have the same dual as (G,τ) ordered by inclusion. Obviously, this poset has always a bottom element, namely the weak topology σ(G,(widehat{G})) . Whether it has also a top element is an open question. We study both quantitative aspects of this poset (its size) and its qualitative aspects, e.g., its chains and anti-chains. Since we are mostly interested in estimates ``from below'', our strategy consists of finding appropriate subgroups (H) of (G) that are easier to handle and show that (mathscr{C} (H)) and (mathscr{C} (G/H)) are large and embed, as a poset, in (mathscr{C}(G,τ)). Important special results are: (i) if (K) is a compact subgroup of a locally quasi-convex group (G), then (mathscr{C}(G)) and (mathscr{C}(G/K)) are quasi-isomorphic (3.15); (ii) if (D) is a discrete abelian group of infinite rank, then (mathscr{C}(D)) is quasi-isomorphic to the poset (mathfrak{F}_D) of filters on D (4.5). Combining both results, we prove that for an LCA (locally compact abelian) group (G ) with an open subgroup of infinite co-rank (this class includes, among others, all non-σ-compact LCA groups), the poset ( mathscr{C} (G) ) is as big as the underlying topological structure of (G,τ) (and set theory) allows. For a metrizable connected compact group (X), the group of null sequences (G=c_0(X)) with the topology of uniform convergence is studied. We prove that (mathscr{C}(G)) is quasi-isomorphic to (mathscr{P}(mathbb{R})) (6.9). Full article

The isomorphism of Karoubi-Villamayor K-groups with smooth K-groups for monoid algebras over quasi stable locally convex algebras is established. We prove that the Quillen K-groups are isomorphic to smooth K-groups for monoid algebras over quasi-stable Frechet algebras having a properly uniformly bounded approximate unit and not necessarily m-convex. Based on these results the K-regularity property for quasi-stable Frechet algebras having a properly uniformly bounded approximate unit is established. Full article