Theorem of Open Mapping (Local Compact Groups)

This article deals with the theorem from the mathematical subfield of locally compact groups. For more sentences of this name see the sentence of the open figure .

The theorem of the open mapping in the mathematical theory of locally compact groups says that a group homomorphism is automatically open in a certain situation.

Terms

A locally compact group is a topological group which, as a topological space, is a locally compact Hausdorff space . Such a space is called σ-compact or countable at infinity if it is the countable union of compact subsets. Group homomorphisms between topological groups are called continuous or open if they are continuous or open as mappings between the topological spaces.

Formulation of the sentence

Let it be a σ-compact, locally compact group and a continuous, surjective group homomorphism on a locally compact group . Then it's open. ${\displaystyle G}$ ${\displaystyle f\colon G\rightarrow H}$ ${\displaystyle H}$ ${\displaystyle f}$

Examples

• The image

${\displaystyle f\colon \mathbb {R} \rightarrow \mathbb {T} :=\{z\in \mathbb {C} \mid |z|=1\},\quad t\mapsto e^{\mathrm {i} t}}$

is a continuous, surjective group homomorphism, if one provides with the addition and with the multiplication as a group structure and they carry the usual topologies. According to the above sentence is open. ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {T} }$ ${\displaystyle f}$

• Let be the group of invertible matrices with the matrix multiplication as the group structure. Then the determinant ${\displaystyle \mathrm {GL} (n,\mathbb {R} )}$

${\displaystyle \det \colon \mathrm {GL} (n,\mathbb {R} )\rightarrow \mathbb {R} ^{*}:=\mathbb {R} \setminus \{0\},\quad A\mapsto \det(A)}$

because of the determinant multiplication theorem, a group homomorphism when looking at the multiplication. The determinant is continuous, because according to the Leibniz formula it is only built up from the sums of products of the matrix components. The determinate is obviously also surjective, because if, then the determinant maps the diagonal matrix with the diagonal on . and are as open subsets of the locally compact spaces and again locally compact, and one easily thinks that it is even σ-compact. With this one can apply the above theorem and obtain that the given determinant mapping is open. ${\displaystyle \mathbb {R} ^{*}}$ ${\displaystyle t\in \mathbb {R} ^{*}}$ ${\displaystyle (t,1,\dotsc ,1)}$ ${\displaystyle t}$ ${\displaystyle \mathrm {GL} (n,\mathbb {R} )}$ ${\displaystyle \mathbb {R} ^{*}}$ ${\displaystyle \mathrm {Mat} (n,\mathbb {R} )\cong \mathbb {R} ^{n^{2}}}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathrm {GL} (n,\mathbb {R} )}$

• Let the additive group with the usual topology and the group of real numbers with the discrete topology . Both are obviously locally compact groups, but are σ-compact, but not. Therefore one cannot apply the theorem to the continuous, surjective group homomorphism and in fact this mapping is not open either. So in the above theorem one cannot do without the requirement of σ-compactness. ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} _{d}}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} _{d}}$ ${\displaystyle \mathrm {id_{\mathbb {R} }} \colon \mathbb {R} _{d}\rightarrow \mathbb {R} }$

Individual evidence

1. ↑ J. Hilgert, K.-H. Neeb: Lie groups and Lie algebras , Vieweg-Verlag (1991), ISBN 978-3-528-06432-7
2. ↑ Markus Stroppel: Locally Compact Groups , European Mathematical Society 2006, ISBN 3-03719-016-7 , sentence 6.19
3. ^ Sidney A. Morris: Pontryagin Duality and the Structure of locally compact abelian groups , Cambridge University Press, ISBN 0-5212-1543-9 , chap. 1, theorem 3
4. ↑ Markus Stroppel: Locally Compact Groups , European Mathematical Society 2006, ISBN 3-03719-016-7 , example 6.20