Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-28T13:32:09.690Z Has data issue: false hasContentIssue false
  • English
  • Français

Studies in harmonic analysis

Published online by Cambridge University Press:  24 October 2008

N. Th. Varopoulos
N. Th. Varopoulos
Affiliation:
Trinity College, Cambridge
Get access Rights & Permissions [Opens in a new window]

Extract

In this paper we shall be mainly concerned with the following three apparently widely differing questions.

(a) What are the possible group topologies on an Abelian group that have a given, fixed continuous character group?

In developing our theory, we are very strongly motivated by the duality theory of linear topological spaces and in particular by Mackey's theorem of that theory. This important result gives a complete characterization of all locally convex topologies on a linear space that have a given, fixed, separating dual space. The analogue of Mackey's theorem for groups, together with related results, is examined in sections 1 and 2 of part 2 of the paper.

(b) What are the properties of topological groups that are denumerable inductive limits of locally compact groups? (See section 1 of part 1 of the paper for definitions.)

Our aim here is to extend results known for locally compact groups to this larger class of groups. The topological study of these groups is carried out in section 3 of part 1 of the paper and the really deep results about their characters are proved in section 5 of part 3 of the paper, as applications of the theory developed in that part of the paper, which is a type of harmonic analysis for these groups.

(c) What are the properties of certain algebras of measures of a locally compact group G, that strictly contain L1(G), and share most of the pleasing properties of L1(G), that is, they do not have any of the pathological features of the full measure algebra M(G) such as the Wiener–Pitt phenomenon or asymmetry?

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 60 , Issue 3 , July 1964 , pp. 465 - 516
Copyright
Copyright © Cambridge Philosophical Society 1964

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1) Bourbaki, N. Topologie générale (Hermann; Paris, 1961).Google Scholar
(2) Bourbaki, N. Espaces vectoriels topologiques (Hermann; Paris, 1953).Google Scholar
(3) Bourbaki, N. Intégration (Hermann; Paris, 1952).Google Scholar
(4) Hewitt, E. The asymmetry of certain algebras of Fourier-Stieltjes transforms. Michigan Math. J. 5 (1958), 149–158.Google Scholar
(5) Kakutani, S. On cardinal numbers related with a compact Abelian group. Proc. Imp. Acad. Tokyo, 19 (1943), 366–372.Google Scholar
(6) Kaplan, S. Extensions of the Pontryagin duality theorem. II. Duke Math. J. 17 (1950), 419–435.CrossRefGoogle Scholar
(7) Kurosh, A. G. The theory of groups, Vol. I (Chelsea; New York, 1955).Google Scholar
(8) Naimark, M. A. Normed rings (Noordhoff; Groningen, 1959).Google Scholar
(9) Northcott, D. G. An introduction to homological algebra (Cambridge, 1960).CrossRefGoogle Scholar
(10) Pontryagin, L. Topological groups (Princeton, 1939).Google Scholar
(11) Rudin, W. Fourier analysis on groups (Interscience; New York, 1962).Google Scholar
(12) Weil, A. L'intégration dans les groups topologiques (Hermann; Paris, 1940).Google Scholar