On group topologies and unitary representations of inductive limits of topological groups and the case of the group of diffeomorphisms

Tatsuuma, Nobuhiko; Shimomura, Hiroaki; Hirai, Takeshi

doi:10.1215/kjm/1250518067

Cited by 25 publications

(57 citation statements)

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“…Thus G is a locally k ω group. The same conclusion holds for countable direct limits of locally compact groups, as considered in [27] and [82]. If each G n is σ-compact, then G is a k ω -group.…”

Section: Example 52mentioning

confidence: 64%

“…We also prove that, for each ascending sequence G 1 ≤ G 2 ≤ · · · of locally k ω groups with continuous inclusion maps, the direct limit topology on G := n∈N G n is locally k ω and makes G a topological group (Proposition 5.4). This result provides a common generalization of known facts concerning direct limits of locally compact groups ( [27,Corollary 3.4], [82,Theorem 2.7]) and abelian k ω -groups [4, Proposition 2.1]. We also formulate a condition ensuring that a final group topology is k ω (Proposition 5.8).…”

Section: Introductionmentioning

confidence: 79%

“…The class of k ω -spaces is quite general and comprises, for example, all countable CW-complexes [73, §5] and countable direct limits of σ-compact locally compact groups (cf. [27], [82]); yet it is restrictive enough to allow a systematic treatment. In the theory of topological groups, k ω -spaces have attracted considerable interest because free topological groups over compact (or k ω -) spaces are k ω -spaces (see [35,Theorem 4], [63,Corollary 1] and [74,Theorem 5.2]), and these are essentially the only examples of free topological groups whose topology is well accessible.…”

Section: Introductionmentioning

confidence: 99%

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Final group topologies, Kac-Moody groups and Pontryagin duality

Glöckner

Gramlich

Hartnick³

2010

Isr. J. Math.

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We study final group topologies and their relations to compactness properties. In particular, we are interested in situations where a colimit or direct limit is locally compact, a kω-space, or locally kω. As a first application, we show that unitary forms of complex Kac-Moody groups can be described as the colimit of an amalgam of subgroups (in the category of Hausdorff topological groups, and the category of kω-groups). Our second application concerns Pontryagin duality theory for the classes of almost metrizable topological abelian groups, resp., locally kω topological abelian groups, which are dual to each other. In particular, we explore the relations between countable projective limits of almost metrizable abelian groups and countable direct limits of locally kω abelian groups.

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Section: Example 52mentioning

confidence: 64%

Section: Introductionmentioning

confidence: 79%

Section: Introductionmentioning

confidence: 99%

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Final group topologies, Kac-Moody groups and Pontryagin duality

Glöckner

Gramlich

Hartnick³

2010

Isr. J. Math.

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“…However, one had to pay a price: Instead of the quite natural topology on Diff(M ) used in Keller's C ∞ c -theory (corresponding to the locally convex topology on D(M, T M )), which makes Diff(M ) a topological group, the convenient approach equips Diff(M ) with a properly finer topology which does not make Diff(M ) a topological group: the group multiplication is discontinuous (cf. [17]). …”

Section: (R T R) ∼ = D(r): the Self-map D(r) → D(r)mentioning

confidence: 99%