Direct sum of topological groups

In mathematics, a topological group G is called the topological direct sum[1] of two subgroups H₁ and H₂ if the map

{\begin{aligned}H_{1}\times H_{2}&\longrightarrow G\\(h_{1},h_{2})&\longmapsto h_{1}h_{2}\end{aligned}}

is a topological isomorphism.

More generally, G is called the direct sum of a finite set of subgroups $H_{i},i=1,\ldots ,n$ of the map

{\begin{aligned}\prod _{i=1}^{n}H_{i}&\longrightarrow G\\(h_{i})_{i\in I}&\longmapsto h_{1}h_{2}\cdots h_{n}\end{aligned}}

Note that if a topological group G is the topological direct sum of the family of subgroups $H_{i}$ then in particular, as an abstract group (without topology) it is also the direct sum (in the usual way) of the family $H_{i}$ .

Topological direct summands

Given a topological group G, we say that a subgroup H is a topological direct summand of G (or that splits topologically from G) if and only if there exist another subgroup K ≤ G such that G is the direct sum of the subgroups H and K.

A the subgroup H is a topological direct summand if and only if the extension of topological groups

0\to H{\stackrel {i}{{}\to {}}}G{\stackrel {\pi }{{}\to {}}}G/H\to 0

splits, where $i$ is the natural inclusion and $\pi$ is the natural projection.

Examples

Suppose that $G$ is a locally compact abelian group that contains the unit circle $\mathbb {T}$ as a subgroup. Then $\mathbb {T}$ is a topological direct summand of G. The same assertion is true for the real numbers $\mathbb {R}$ [2]

References

E. Hewitt and K. A. Ross, Abstract harmonic analysis. Vol. I, second edition, Grundlehren der Mathematischen Wissenschaften, 115, Springer, Berlin, 1979. MR0551496 (81k:43001)
Armacost, David L. The structure of locally compact abelian groups. Monographs and Textbooks in Pure and Applied Mathematics, 68. Marcel Dekker, Inc., New York, 1981. vii+154 pp. ISBN 0-8247-1507-1 MR0637201 (83h:22010)

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[1] E. Hewitt and K. A. Ross, Abstract harmonic analysis. Vol. I, second edition, Grundlehren der Mathematischen Wissenschaften, 115, Springer, Berlin, 1979. MR0551496 (81k:43001)

[2] Armacost, David L. The structure of locally compact abelian groups. Monographs and Textbooks in Pure and Applied Mathematics, 68. Marcel Dekker, Inc., New York, 1981. vii+154 pp. ISBN 0-8247-1507-1 MR0637201 (83h:22010)