Modular function (harmonic analysis)
The modular function is a term from harmonic analysis , i.e. from the theory of locally compact groups . The modular function measures a left-right asymmetry of the group.
definition
Let it be a locally compact group. Then there is known to be a linksinvariantes Haar measure on . Left invariance means that for all and all Borel sets . In general, it does not follow from this that is also right invariant, that is, it can absolutely apply.
For solid , the figure is also a left-invariant Haar's measure, as can easily be confirmed. Since this is uniquely determined except for a constant, there is a number with , that is, for all measurable ones .
In this way, a mapping is obtained that proves to be independent of the choice of the left-invariant Haar measure and is a continuous homomorphism of in the multiplicative group . is the modular function of
Unimodular groups
Groups for which the modular function is the same as the constant function for all are called unimodular . These are exactly those groups for which a left-invariant Haar's measure is also right-invariant. Three important types of locally compact groups are automatically unimodular:
- Commutative locally compact groups are unimodular, because due to the commutativity, left-invariant measures are naturally also right-invariant.
- Compact groups are bound one, because the image of the modular function has a compact subgroup in his, and there is only the question.
- Discrete groups are unimodular, because the multiples of the counting measure are exactly the left and right invariant Haarsche measures.
An example of a unimodular, locally compact group that does not fall under any of these three types is the general linear group . A left and right invariant measure is through
given, where the Lebesgue measure is on .
example
Here we give an example of a non-trivial modular function. Let it be the locally compact group of all matrices with . A left-invariant Haar's measure is given by, a right-invariant one by . This results in .
Calculation rules
Let it be a locally compact group with left invariant Haar's measure . For a function, let the so-called translation of um .
If the characteristic function is the Borel set , then and therefore according to the construction of the modular function
.
With the usual mass theoretic inferences we get for every -integratable function :
.
The modular function also occurs if one integrates via inverted arguments. For -integratable functions on applies
.
Finally, the modular function occurs in the definition of the involution on convolutional algebra . In the space above, define for function
.
It is only defined almost everywhere , namely where the integral exists, and the dash stands for the complex conjugation . With the so-called convolution product defined by and the mapping , a Banach algebra with isometric involution becomes . The study of this Banach algebra is an important tool in harmonic analysis .
Individual evidence
- ^ Donald L. Cohn: Measure Theory. Birkhäuser, Boston MA 1980, ISBN 3-7643-3003-1 , set 9.3.4.
- ^ Donald L. Cohn: Measure Theory. Birkhäuser, Boston MA 1980, ISBN 3-7643-3003-1 , chapter 9.3, exercise 2
- ^ Donald L. Cohn: Measure Theory. Birkhäuser, Boston MA 1980, ISBN 3-7643-3003-1 , text after sentence 9.3.3.
- ↑ Lynn H. Loomis : An Introduction to Abstract Harmonic Analysis. D. van Nostrand Co., Princeton NJ et al. 1953, § 30B.
- ^ Jacques Dixmier : C * -algebras (= North-Holland Mathematical Library. Vol. 15). North Holland Publishing Company, Amsterdam et al. 1977, ISBN 0-7204-2450-X , Chapter 13.2.