Book contents
- Frontmatter
- Contents
- Preface
- 1 Introduction to topological groups
- 2 Subgroups and quotient groups of Rn
- 3 Uniform spaces and dual groups
- 4 Introduction to the Pontryagin-van Kampen duality theorem
- 5 Duality for compact and discrete groups
- 6 The duality theorem and the principal structure theorem
- 7 Consequences of the duality theorem
- 8 Locally Euclidean and NSS-groups
- 9 Non-abelian groups
- References
- Index of terms
- Index of Exercises, propositions and theorems
9 - Non-abelian groups
Published online by Cambridge University Press: 11 November 2009
- Frontmatter
- Contents
- Preface
- 1 Introduction to topological groups
- 2 Subgroups and quotient groups of Rn
- 3 Uniform spaces and dual groups
- 4 Introduction to the Pontryagin-van Kampen duality theorem
- 5 Duality for compact and discrete groups
- 6 The duality theorem and the principal structure theorem
- 7 Consequences of the duality theorem
- 8 Locally Euclidean and NSS-groups
- 9 Non-abelian groups
- References
- Index of terms
- Index of Exercises, propositions and theorems
Summary
In this chapter we make a few remarks about non-abelian locally compact Hausdorff groups.
For compact Hausdorff (not necessarily abelian) groups there is a duality theory due to M.G. Krein and T. Tannaka. The dual object of a compact Hausdorff group G is not another topological group, as in the abelian case, but rather the class of continuous finite-dimensional unitary representations of G (or a Krein algebra). For full details, see E. Hewitt and K.A. Ross, Abstract harmonic analysis Vol.11.
Let H be a complex vector space and T(H) the group
of all one-one linear transformations of H onto itself.
A representation of a group G is a map of G
into T(H) such that, for each x and y
in G, with the identity operator. A representation U of a topological group is said to be a continuous irreducible unitary representation if (a) H is a Hilbert space, (b) every transformation is unitary on
H, (c) for every and η in H, the function
of G into the complex numbers is continuous, and
(d) there are no proper closed subspaces of H carried into
themselves by every.
The central theorem in representation theory of topological groups is due to I.M. Gelfand and D.A. Raikov.
Theorem (Gelfand-Raikov). Every locally compact Hausdorff group G has enough continuous irreducible unitary representations to separate points.
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- Publisher: Cambridge University PressPrint publication year: 1977