Contributions to the duality theory of abelian topological groups and to the theory of nuclear groups

Außenhofer Lydia

GDML_Books, (1999), p.

Résumé

Abstract For a topological group G, the group G* of continuous homomorphisms (characters) into :=z∈ℂ: |z| = 1 endowed with the compact-open topology is called the character group of G and G is named ( Pontryagin) reflexive if the canonical homomorphism $α_G:G → G**$, x ↦ (χ ↦ χ(x)), is a topological isomorphism. A comprehensive exposition of duality theory is given here. In particular, settings closely related to the theory of vector spaces (like local quasi-convexity and the corresponding hull) are studied and their relevance is pointed out. This is followed by an investigation of Pontryagin reflexivity of locally convex vector spaces, which generalizes the well known fact that every Banach space is a reflexive group. However, the spaces $L^p([0,1])$ (for p > 1) contain proper closed subgroups which are not reflexive and have (topologically) the same character group as the whole space. On the other hand, every character group can be embedded into a group of the form C(X,). It is proved that for every hemicompact k-space X (in particular, for every character group of an abelian metrizable group), this group is reflexive. In the second part a self-contained introduction to the theory of nuclear groups (which has been introduced by W. Banaszczyk in [9]) is given. It is shown that the completion of a nuclear group is again nuclear and that the α corresponding to a complete nuclear group is surjective. In particular, every Čech-complete nuclear group is (strongly) reflexive. At the end, a simplified proof of the Bochner Theorem for nuclear groups is given.

CONTENTS Introduction...........................................................................................7 Notation...............................................................................................10 1. Auxiliary results in topology..............................................................11 2. Auxiliary results for topological groups............................................17 3. Elementary properties of homomorphism groups............................21 4. Homomorphism groups of abelian metrizable groups......................23 5. Some results in duality theory.........................................................25 6. Locally quasi-convex groups...........................................................30 7. Properties of the quasi-convex hull.................................................36 8. Reflexivity of locally convex vector spaces......................................40 9. Locally convex vector groups..........................................................46 10. Two representations of locally quasi-convex groups.....................48 11. The character groups of $L^p_ℤ([0,1])$ and $L^p([0,1])$............53 12. Free abelian topological groups...................................................58 13. C(K,) for compact K....................................................................63 14. The group C(X,)..........................................................................69 15. Duality theory for free abelian topological groups.........................71 16. A short survey of the theory of nuclear groups.............................73 17. Ellipsoids.......................................................................................73 18. Properties of the Kolmogorov diameter.........................................75 19. Gaussian-like measures................................................................86 20. Nuclear groups..............................................................................93 21. An embedding theorem for nuclear groups.................................104 22. The Bochner Theorem for nuclear groups..................................105 References........................................................................................111

1991 Mathematics Subject Classification: 11H06, 22Axx, 22B99, 43A40, 43A35, 46Axx, 52C07, 52A40, 54Dxx, 54E15, 60B15.

Détail

Zbl 0905.22001

EUDML-ID : urn:eudml:doc:275826