J. Aust. Math. Soc. 83 (2007), no. 1, pp. 55–77.
An Open Mapping theorem for pro-Lie groups
Karl H. Hofmann Sidney A. Morris†
Fachbereich Mathematik
Darmstadt University of Technology
Schlossgartenstr. 7
D-64289 Darmstadt
Germany
hofmann@mathematik.tu-darmstadt.de		 School of Information Technology and Mathematical Sciences
University of Ballarat
P.O. Box 663
Ballarat Victoria 3353
Australia
s.morris@ballarat.edu.au
Received 4 July 2005; revised 10 April 2006
Communicated by G. Willis

Abstract

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.

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2000 Mathematics Subject Classification: primary 22A05, 22E65; secondary 46A30
(Metadata: XML, RSS, BibTeX) MathSciNet: MR2354??? Z'blatt-MATH: pre05231332
†indicates author for correspondence

References

  1. N. Bourbaki, Groupes et algèbres de Lie, chapters 2–3 (Hermann, Paris, 1972). MR573068
  2. N. Bourbaki, Topologie générale, chapters 5–10 (Hermann, Paris, 1974). MR1726872
  3. E. Hewitt and K. A. Ross, Abstract Harmonic Analysis. Vol. I Structure of topological groups. Integration theory, group representations (Academic Press, Inc., Publishers, New York, 1963). MR156915
  4. K. H. Hofmann, ‘On a category of topological groups suitable for a structure theory of locally compact groups’, Topology Proceedings 26 (2001–2002), 651–665. MR2032841
  5. K. H. Hofmann and S. A. Morris, The Structure of Compact Groups (De Gruyter Berlin, 1998). MR1646190
  6. K. H. Hofmann and S. A. Morris, ‘Projective limits of finite dimensional Lie groups’, Proc. London Math. Soc. 87 (2003), 647–676. MR2005878
  7. K. H. Hofmann and S. A. Morris, ‘The structure of abelian pro-Lie groups’, Math. Z. 248 (2004), 867–891. MR2103546
  8. K. H. Hofmann and S. A. Morris, ‘Sophus Lie's third fundamental theorem and the adjoint functor theorem’, J. Group Theory 8 (2005), 115–133. MR2115603
  9. K. H. Hofmann and S. A. Morris, The Lie Theory of Connected Pro-Lie Groups- the Structure of Pro-Lie Algebra, Pro-Lie Groups and Locally Compact Groups (EMS Publishing House, Zürich, 2007). MR2337107
  10. K. H. Hofmann, S. A. Morris and D. Poguntke, ‘The exponential function of locally connected compact abelian groups’, Forum Math. 16 (2003), 1–16. MR2034540
  11. F. Burton Jones, ‘Connected and disconnected plane sets and the functional equation f(x+y)=f(x)+f(y)’, Bull. Amer. Math. Soc. 48 (1942), 115–120. MR5906
  12. Sh. Koshi and M. Takesaki, ‘An open mapping theorem on homogeneous spaces’, J. Aust. Math. Soc., Ser. A. 53 (1992), 51–54. MR1164775
  13. D. Montgomery and L. Zippin, Topological Transformation Groups (Interscience Publishers, New York, 1955). MR73104
  14. W. Roelcke and S. Dierolf, Uniform Structures on Topological Groups and their Quotients (McGraw-Hill, New York, 1981). MR644485
  15. H. Yamabe, ‘Generalization of a theorem of Gleason’, Ann. of Math. 58 (1953), 351–365. MR58607
  16. H. Yamabe, ‘On the conjecture of Iwasawa and Gleason’, Ann. of Math. 58 (1953), 48–54. MR54613