43 research outputs found
Locally Quasi-Convex Compatible Topologies on a Topological Group
(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)
“Varopoulos paradigm”: Mackey property versus metrizability in topological groups.
The class of all locally quasi-convex (lqc) abelian groups contains all locally convex vector spaces (lcs) considered as topological groups. Therefore it is natural to extend classical properties of locally convex spaces to this larger class of abelian topological groups. In the present paper we consider the following well known property of lcs: “A metrizable locally convex space carries its Mackey topology ”. This claim cannot be extended to lqc-groups in the natural way, as we have recently proved with other coauthors (Außenhofer and de la Barrera Mayoral in J Pure Appl Algebra 216(6):1340–1347, 2012; Díaz Nieto and Martín Peinador in Descriptive Topology and Functional Analysis, Springer Proceedings in Mathematics and Statistics, Vol 80 doi:10.1007/978-3-319-05224-3_7, 2014; Dikranjan et al. in Forum Math 26:723–757, 2014). We say that an abelian group G satisfies the Varopoulos paradigm (VP) if any metrizable locally quasi-convex topology on G is the Mackey topology. In the present paper we prove that in any unbounded group there exists a lqc metrizable topology that is not Mackey. This statement (Theorem C) allows us to show that the class of groups satisfying VP coincides with the class of finite exponent groups. Thus, a property of topological nature characterizes an algebraic feature of abelian groups
Precompact noncompact reflexive abelian groups
We present a series of examples of precompact, noncompact, reflexive
topological Abelian groups. Some of them are pseudocompact or even countably
compact, but we show that there exist precompact non-pseudocompact reflexive
groups as well. It is also proved that every pseudocompact Abelian group is a
quotient of a reflexive pseudocompact group with respect to a closed reflexive
pseudocompact subgroup
Quasi-convex density and determining subgroups of compact abelian groups
For an abelian topological group G let G^* denote the dual group of all
continuous characters endowed with the compact open topology. Given a closed
subset X of an infinite compact abelian group G such that w(X) < w(G) and an
open neighbourhood U of 0 in the circle group, we show that the set of all
characters which send X into U has the same size as G^*. (Here, w(G) denotes
the weight of G.) A subgroup D of G determines G if the restriction
homomorphism G^* --> D^* is an isomorphism between G^* and D^*. We prove that
w(G) = min {|D|: D is a subgroup of G that determines G} for every infinite
compact abelian group G. In particular, an infinite compact abelian group
determined by a countable subgroup is metrizable. This gives a negative answer
to questions of Comfort, Hernandez, Macario, Raczkowski and Trigos-Arrieta. As
an application, we furnish a short elementary proof of the result from [13]
that a compact abelian group G is metrizable provided that every dense subgroup
of G determines G.Comment: 11 pages. The proof of Lemma 3.6 (from version 2) has been
significantly simplified and shorten. 2 lemmas and 4 references have been
added. The order of the material has been substantially changed as wel
Differentiable mappings on products with different degrees of differentiability in the two factors
We develop differential calculus of -mappings on products of locally
convex spaces and prove exponential laws for such mappings. As an application,
we consider differential equations in Banach spaces depending on a parameter in
a locally convex space. Under suitable assumptions, the associated flows are
mappings of class .Comment: 42 Pages, Latex, v2: Corrected several misprints and typos. The
results remain unchange
Direct limits of infinite-dimensional Lie groups compared to direct limits in related categories
Let G be a Lie group which is the union of an ascending sequence of Lie
groups G_n (all of which may be infinite-dimensional). We study the question
when G is the direct limit of the G_n's in the category of Lie groups,
topological groups, smooth manifolds, resp., topological spaces. Full answers
are obtained for G the group Diff_c(M) of compactly supported smooth
diffeomorphisms of a sigma-compact smooth manifold M, and for test function
groups C^infty_c(M,H) of compactly supported smooth maps with values in a
finite-dimensional Lie group H. We also discuss the cases where G is a direct
limit of unit groups of Banach algebras, a Lie group of germs of Lie
group-valued analytic maps, or a weak direct product of Lie groups.Comment: extended preprint version, 66 page
The character of topological groups, via bounded systems, Pontryagin--van Kampen duality and pcf theory
The Birkhoff--Kakutani Theorem asserts that a topological group is metrizable
if and only if it has countable character. We develop and apply tools for the
estimation of the character for a wide class of nonmetrizable topological
groups.
We consider abelian groups whose topology is determined by a countable
cofinal family of compact sets. These are the closed subgroups of
Pontryagin--van Kampen duals of \emph{metrizable} abelian groups, or
equivalently, complete abelian groups whose dual is metrizable. By
investigating these connections, we show that also in these cases, the
character can be estimated, and that it is determined by the weights of the
\emph{compact} subsets of the group, or of quotients of the group by compact
subgroups. It follows, for example, that the density and the local density of
an abelian metrizable group determine the character of its dual group. Our main
result applies to the more general case of closed subgroups of Pontryagin--van
Kampen duals of abelian \v{C}ech-complete groups.
In the special case of free abelian topological groups, our results extend a
number of results of Nickolas and Tkachenko, which were proved using
combinatorial methods.
In order to obtain concrete estimations, we establish a natural bridge
between the studied concepts and pcf theory, that allows the direct application
of several major results from that theory. We include an introduction to these
results and their use.Comment: Minor corrections. Final version before journal editin
The Lie algebra of a nuclear group
The Lie algebra of a nuclear group is a locally convex nuclear vector space