Locally quasi-convex compatible topologies on locally compact abelian groups
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Mathematische Zeitschrift
Locally quasi-convex compatible topologies on locally compact abelian groups Lydia Außenhofer1 · Dikran Dikranjan2 Received: 30 October 2018 / Accepted: 8 October 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract Let G be a locally compact abelian group (LCA group). Our aim is to study the poset C (G) of all locally quasi-convex topologies on G that are compatible (i.e., have the same dual as G) ordered by inclusion. This poset has a bottom element, namely the Bohr topology σ (G, G) and a top element, namely the original topology. We shall be interested in both quantitative aspects of this poset (its size) and its qualitative aspects (its “width”, namely its anti-chain number). Since we are mostly interested in estimates “from below”, our strategy will be to embed well known posets such as the poset of free filters or the poset ([ω]ω , ⊆∗ ) in C (G). We show that for every LCA group G the set of compatible topologies C (G) is order-isomorphic to C (D) for a suitable discrete group D, e.g. C (Rn ) ∼ = C (Zn ) for all n ∈ N. If the rank of D is infinite, it was already shown in Außenhofer et al. (Axioms 4:436–458, 2019) that C (D) is as big as possible. In this paper we prove that C (R) ∼ = C (Z) and C (Z( p ∞ )) have width and hence size at least c and that they contain chains of length c. This yields that for any non-compact LCA group G the set C (G) has width ≥ c and chains of length ≥ c. Further, we characterize the discrete groups D such that C (D) may fail to have the maximal possible cardinality and width (they are at most countably many, up to isomorphism). Keywords Ordered set · Lattice · Locally quasi-convex topology · Compatible topology · Quasi-convex sequence · Quasi-isomorphic posets · Free filters · Mackey groups · LCA group Mathematics Subject Classification Primary 20K45 · 22A05 · 22B05 · 43A40 · 54H11; Secondary 03G10 · 06A11 · 46A03 · 46E05 · 20K25
Respectfully dedicated to the 150-birthday of Felix Hausdorff. The authors thank IMI for supporting a visit to U.C.M. along the last term of 2009, which has allowed them to work in this topic. L. Außenhofer is deeply grateful to the University of Udine for the invitations in 2016 and 2018 to the Department of Mathematical, Computer and Physical Sciences which enabled the authors to work on this topic. D. Dikranjan was supported by grant PSD-2015-2017-DIMA-PRID- 2017-DIKRANJAN PSD-2015-2017-DIMA of Udine University. This author gratefully acknowledges the warm hospitality at the Faculty of Computer Sciences and Mathematics of the University of Passau during his visit in February 2017.
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Lydia Außenhofer [email protected]
Extended author information available on the last page of the article
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L. Außenhofer, D. Dikranjan
1 Introduction Varopoulos posed the question which group topologies on an abelian group G have a given continuous character group H , and called them compatible topologies for the duality (G, H ), [36]. As the author explains, the question is motivated by the Mac
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