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{align} H_1\times H_1 &\longrightarrow G \\ (h_1,h_2) &\longmapsto h_1 h_2 \end{align}

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{align} \prod^n_{i=1} H_i&\longrightarrow G \\ (h_i)_{i\in I} &\longmapsto h_1 h_2 \cdots h_n \end{align}

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 form 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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