The Structure of Compact Groups

Abstract

The theme of this book is the Structure Theory of compact groups. It contains a completely selfcontained introduction to linear Lie groups and a substantial body of material on compact Lie groups. The authors' approach is distinctive in so far as they define a linear Lie group as a particular subgroup of the multiplicative group of a Banach algebra. Compact Lie groups are recognized at an early stage as being linear Lie groups. This approach avoids the use of machinery on manifolds. The text is written in a style to make it accessible to the beginning graduate student with a basic knowledge in analysis, algebra, and topology. At the same time the expert will find it an excellent and rich source of information on the general structure theory of compact groups.

Loops as sections in compact Lie groups

“…Since any 1-dimensional subgroup of K 1 is contained in a 3-dimensional subgroup of K 1 locally isomorphic to SO 3 (R) (cf. [5], Propositions 6.45 and 6.46) and by Corollary 5 any 3-dimensional loop homeomorphic to a cover of SO 3 (R) is a group, besides a homomorphism with finite kernel we get that x → x n is either an isomorphism or an anti-isomorphism of SO 3 (R). Hence one has n = 1 or −1.…”

“…Since the 3-dimensional subgroups of H 1 covers H 1 (cf. [5], Propositions 6.45 and 6.46) for the continuous section σ one has σ(H 1 ) = (H 1 , 1).…”

The weights of closed subgroups of a locally compact group

“…We refer to a cardinal invariant for compact groups G which is one of several alternatives to the weight w.G/, namely, the so-called generating degree s.G/ (see [9,Definition 12.15]). The definition relies on the Suitable Set Theorem, loc.…”

“…cit. Theorem 12.11, which in turn invokes the so-called Countable Layer Theorem (see [8] or [9,Theorem 9.91]). Indeed recall that in a compact group G a subset S is called suitable iff it does not contain 1, is closed and discrete in G n ¹1º, and satisfies G D hSi.…”

“…By Proposition 12.28 of [9], for an infinite profinite, that is, compact totally disconnected group we have…”