Modulus of an automorphism
A positive real number associated to an automorphism of a locally compact group. Let be such a group and let
be an automorphism of
, regarded as a topological group. Then the modulus of
is defined by
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where is left-invariant Haar measure on
and
is any compact subset of
with positive measure (indeed,
does not depend on
). If
is compact or discrete, then
, since for a compact group one can put
, and for a discrete group one can take
, where
is the identity element of
.
If and
are two automorphism of
, then
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If is a topological group which acts continuously on
by automorphisms, then the associated homomorphism
defines a continuous homomorphism
, where
is the multiplicative group of positive real numbers. In particular, if
and
, then
is a continuous homomorphism. This homomorphism is trivial if and only if the left-invariant Haar measure on
is simultaneously right invariant. Groups satisfying the latter condition are called unimodular.
If is a locally compact skew-field, then each non-zero element
defines an automorphism
of the additive group of
via multiplication by
. The function
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) |
Modulus of an automorphism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Modulus_of_an_automorphism&oldid=47878