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A positive real number associated to an automorphism of a locally compact group. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064550/m0645501.png" /> be such a group and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064550/m0645502.png" /> be an automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064550/m0645503.png" />, regarded as a topological group. Then the modulus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064550/m0645504.png" /> is defined by
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064550/m0645505.png" /></td> </tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064550/m0645506.png" /> is left-invariant [[Haar measure|Haar measure]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064550/m0645507.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064550/m0645508.png" /> is any compact subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064550/m0645509.png" /> with positive measure (indeed, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064550/m06455010.png" /> does not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064550/m06455011.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064550/m06455012.png" /> is compact or discrete, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064550/m06455013.png" />, since for a compact group one can put <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064550/m06455014.png" />, and for a discrete group one can take <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064550/m06455015.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064550/m06455016.png" /> is the identity element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064550/m06455017.png" />.
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A positive real number associated to an automorphism of a locally compact group. Let  $  G $
 +
be such a group and let  $  \alpha $
 +
be an automorphism of  $  G $,  
 +
regarded as a topological group. Then the modulus of $  \alpha $
 +
is defined by
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064550/m06455018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064550/m06455019.png" /> are two automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064550/m06455020.png" />, then
+
$$
 +
\mathop{\rm mod} _ {G} ( \alpha )  = \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064550/m06455021.png" /></td> </tr></table>
+
\frac{\mu ( \alpha S ) }{\mu ( S) }
 +
,
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064550/m06455022.png" /> is a topological group which acts continuously on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064550/m06455023.png" /> by automorphisms, then the associated homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064550/m06455024.png" /> defines a continuous homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064550/m06455025.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064550/m06455026.png" /> is the multiplicative group of positive real numbers. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064550/m06455027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064550/m06455028.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064550/m06455029.png" /> is a continuous homomorphism. This homomorphism is trivial if and only if the left-invariant Haar measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064550/m06455030.png" /> is simultaneously right invariant. Groups satisfying the latter condition are called unimodular.
+
where  $  \mu $
 +
is left-invariant [[Haar measure|Haar measure]] on $  G $
 +
and  $  S $
 +
is any compact subset of $  G $
 +
with positive measure (indeed, $  \mathop{\rm mod} _ {G} ( \alpha ) $
 +
does not depend on  $  S $).  
 +
If  $  G $
 +
is compact or discrete, then $  \mathop{\rm mod} _ {G} ( \alpha ) \equiv 1 $,
 +
since for a compact group one can put  $  S = G $,
 +
and for a discrete group one can take  $  S = \{ 1 \} $,
 +
where  $  1 $
 +
is the identity element of  $  G $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064550/m06455031.png" /> is a locally compact skew-field, then each non-zero element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064550/m06455032.png" /> defines an automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064550/m06455033.png" /> of the additive group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064550/m06455034.png" /> via multiplication by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064550/m06455035.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064550/m06455036.png" /> is used in the study of the structure of locally compact skew-fields.
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If $  \alpha $
 +
and  $  \beta $
 +
are two automorphism of  $  G $,
 +
then
 +
 
 +
$$
 +
\mathop{\rm mod} _ {G} ( \alpha \cdot \beta )  = \
 +
\mathop{\rm mod} _ {G} ( \alpha ) \
 +
\mathop{\rm mod} _ {G} ( \beta ) .
 +
$$
 +
 
 +
If  $  \Gamma $
 +
is a topological group which acts continuously on  $  G $
 +
by automorphisms, then the associated homomorphism  $  \pi :  \Gamma \rightarrow  \mathop{\rm Aut}  G $
 +
defines a continuous homomorphism  $  \mathop{\rm mod} _ {G} \circ \pi :  \Gamma \rightarrow \mathbf R _ {+}  ^ {*} $,
 +
where  $  \mathbf R _ {+}  ^ {*} $
 +
is the multiplicative group of positive real numbers. In particular, if  $  \Gamma = G $
 +
and  $  \pi ( g) ( x) = g x g  ^ {-} 1 $,
 +
then  $  \pi \circ  \mathop{\rm mod} _ {G} : G \rightarrow \mathbf R _ {+}  ^ {*} $
 +
is a continuous homomorphism. This homomorphism is trivial if and only if the left-invariant Haar measure on  $  G $
 +
is simultaneously right invariant. Groups satisfying the latter condition are called unimodular.
 +
 
 +
If  $  K $
 +
is a locally compact skew-field, then each non-zero element $  a \in K $
 +
defines an automorphism $  \mu ( a) $
 +
of the additive group of $  K $
 +
via multiplication by $  a $.  
 +
The function $  \mathop{\rm mod} _ {K} \circ \mu :  K \setminus  \{ 0 \} \rightarrow \mathbf R _ {+}  ^ {*} $
 +
is used in the study of the structure of locally compact skew-fields.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Integration" , Addison-Wesley  (1975)  pp. Chapt.6;7;8  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Weil,  "l'Intégration dans les groupes topologiques et ses applications" , Hermann  (1940)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Weil,  "Basic number theory" , Springer  (1974)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Integration" , Addison-Wesley  (1975)  pp. Chapt.6;7;8  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Weil,  "l'Intégration dans les groupes topologiques et ses applications" , Hermann  (1940)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Weil,  "Basic number theory" , Springer  (1974)</TD></TR></table>

Latest revision as of 08:01, 6 June 2020


A positive real number associated to an automorphism of a locally compact group. Let $ G $ be such a group and let $ \alpha $ be an automorphism of $ G $, regarded as a topological group. Then the modulus of $ \alpha $ is defined by

$$ \mathop{\rm mod} _ {G} ( \alpha ) = \ \frac{\mu ( \alpha S ) }{\mu ( S) } , $$

where $ \mu $ is left-invariant Haar measure on $ G $ and $ S $ is any compact subset of $ G $ with positive measure (indeed, $ \mathop{\rm mod} _ {G} ( \alpha ) $ does not depend on $ S $). If $ G $ is compact or discrete, then $ \mathop{\rm mod} _ {G} ( \alpha ) \equiv 1 $, since for a compact group one can put $ S = G $, and for a discrete group one can take $ S = \{ 1 \} $, where $ 1 $ is the identity element of $ G $.

If $ \alpha $ and $ \beta $ are two automorphism of $ G $, then

$$ \mathop{\rm mod} _ {G} ( \alpha \cdot \beta ) = \ \mathop{\rm mod} _ {G} ( \alpha ) \ \mathop{\rm mod} _ {G} ( \beta ) . $$

If $ \Gamma $ is a topological group which acts continuously on $ G $ by automorphisms, then the associated homomorphism $ \pi : \Gamma \rightarrow \mathop{\rm Aut} G $ defines a continuous homomorphism $ \mathop{\rm mod} _ {G} \circ \pi : \Gamma \rightarrow \mathbf R _ {+} ^ {*} $, where $ \mathbf R _ {+} ^ {*} $ is the multiplicative group of positive real numbers. In particular, if $ \Gamma = G $ and $ \pi ( g) ( x) = g x g ^ {-} 1 $, then $ \pi \circ \mathop{\rm mod} _ {G} : G \rightarrow \mathbf R _ {+} ^ {*} $ is a continuous homomorphism. This homomorphism is trivial if and only if the left-invariant Haar measure on $ G $ is simultaneously right invariant. Groups satisfying the latter condition are called unimodular.

If $ K $ is a locally compact skew-field, then each non-zero element $ a \in K $ defines an automorphism $ \mu ( a) $ of the additive group of $ K $ via multiplication by $ a $. The function $ \mathop{\rm mod} _ {K} \circ \mu : K \setminus \{ 0 \} \rightarrow \mathbf R _ {+} ^ {*} $ is used in the study of the structure of locally compact skew-fields.

References

[1] N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French)
[2] A. Weil, "l'Intégration dans les groupes topologiques et ses applications" , Hermann (1940)
[3] A. Weil, "Basic number theory" , Springer (1974)
How to Cite This Entry:
Modulus of an automorphism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Modulus_of_an_automorphism&oldid=47878
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article