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An Open Mapping Theorem For Pro-Lie Groups

Part of: Topological linear spaces and related structures Lie groups Topological and differentiable algebraic systems

Published online by Cambridge University Press:  09 April 2009

Karl H. Hofmann  and
Sidney A. Morris
Karl H. Hofmann
Affiliation:
Fachbereich MathematikDarmstadt University of TechnologySchlossgartenstr. 7 D-64289 DarmstadtGermanyhofmann@mathematik.tu-darmstadt.de
Sidney A. Morris
Affiliation:
School of Information Technology and Mathematical Sciences University of BallaratP.O. Box 663 Ballarat Victoria 3353Australias.morris@ballarat.edu.au
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Abstract

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A pro-Lie group is a projective limit of finite dimensional Lie groups. It is proved that a surjective continuous group homomorphism between connected pro-Lie groups is open. In fact this remains true for almost connected pro-Lie groups where a topological group is called almost connected if the factor group modulo the identity component is compact. As consequences we get a Closed Graph Theorem and the validity of the Second Isomorphism Theorem for pro-Lie groups in the almost connected context.

MSC classification

Secondary: 22A05: Structure of general topological groups 22E65: Infinite-dimensional Lie groups and their Lie algebras 46A30: Open mapping and closed graph theorems; completeness (including $B$-, $B_r$-completeness)

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 83 , Issue 1 , August 2007 , pp. 55 - 78
Copyright
Copyright © Australian Mathematical Society 2007

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