# 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. ${\ displaystyle G}$ ${\ displaystyle \ mu}$${\ displaystyle G}$${\ displaystyle \ mu (tA) = \ mu (A)}$${\ displaystyle t \ in G}$ ${\ displaystyle A \ subset G}$${\ displaystyle \ mu}$${\ displaystyle \ mu (At) \ not = \ mu (A)}$

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 . ${\ displaystyle t \ in G}$${\ displaystyle \ mu _ {t}: \, A \ mapsto \ mu (At)}$${\ displaystyle \ Delta _ {G} (t) \ in \ mathbb {R} ^ {+}}$${\ displaystyle \ mu _ {t} \, = \, \ Delta _ {G} (t) \ mu}$${\ displaystyle \ mu (At) \, = \, \ Delta _ {G} (t) \ mu (A)}$${\ displaystyle A \ subset G}$

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${\ displaystyle \ Delta _ {G}: G \ rightarrow \ mathbb {R} ^ {+}}$${\ displaystyle \ mu}$ ${\ displaystyle G}$${\ displaystyle \ mathbb {R} ^ {+}}$${\ displaystyle \ Delta _ {G}}$${\ displaystyle G}$

## 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: ${\ displaystyle \ Delta _ {G} (t) = 1}$${\ displaystyle t \ in G}$

• 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.${\ displaystyle \ mathbb {R} ^ {+}}$${\ displaystyle \ {1 \}}$
• 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 ${\ displaystyle GL (n, \ mathbb {R})}$

${\ displaystyle \ mu (A) = \ int _ {A} {\ frac {1} {| \ det (u) |}} \, \ mathrm {d} \ lambda (u)}$

given, where the Lebesgue measure is on . ${\ displaystyle \ lambda}$${\ displaystyle \ mathbb {R} ^ {n ^ {2}}}$

## 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 . ${\ displaystyle G}$${\ displaystyle 2 \ times 2}$${\ displaystyle {\ begin {pmatrix} a & b \\ 0 & 1 \ end {pmatrix}}}$${\ displaystyle a, b \ in \ mathbb {R}, a> 0}$${\ displaystyle \ mu (A) = \ int _ {\ mathbb {R}} \ int _ {\ mathbb {R} ^ {+}} {\ frac {1} {a ^ {2}}} \, \ mathrm {d} a \ mathrm {d} b}$${\ displaystyle \ nu (A) = \ int _ {\ mathbb {R}} \ int _ {\ mathbb {R} ^ {+}} {\ frac {1} {a}} \, \ mathrm {d} a \ mathrm {d} b}$${\ displaystyle \ Delta _ {G} ({\ begin {pmatrix} a & b \\ 0 & 1 \ end {pmatrix}}) \, = \, {\ frac {1} {a}}}$

## 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 . ${\ displaystyle G}$${\ displaystyle \ mu}$${\ displaystyle f: G \ rightarrow R}$${\ displaystyle f_ {s} (t) \,: = \, f (ts ^ {- 1})}$${\ displaystyle f}$${\ displaystyle s}$

If the characteristic function is the Borel set , then and therefore according to the construction of the modular function ${\ displaystyle \ chi _ {A}}$${\ displaystyle A}$${\ displaystyle (\ chi _ {A}) _ {s} = \ chi _ {As}}$

${\ displaystyle \ int (\ chi _ {A}) _ {s} (t) \ mathrm {d} \ mu (t) = \ int \ chi _ {As} (t) \ mathrm {d} \ mu ( t) = \ mu (As) = \ Delta _ {G} (s) \ mu (A) = \ Delta _ {G} (s) \ int \ chi _ {A} (t) \ mathrm {d} \ mu (t)}$.

With the usual mass theoretic inferences we get for every -integratable function : ${\ displaystyle \ mu}$${\ displaystyle f}$

${\ displaystyle \ int f_ {s} (t) \ mathrm {d} \ mu (t) = \ Delta _ {G} (s) \ int f (t) \ mathrm {d} \ mu (t)}$.

The modular function also occurs if one integrates via inverted arguments. For -integratable functions on applies ${\ displaystyle \ mu}$${\ displaystyle f}$${\ displaystyle G}$

${\ displaystyle \ int f (t ^ {- 1}) \ Delta _ {G} (t ^ {- 1}) \ mathrm {d} \ mu (t) \, = \, \ int f (t) \ mathrm {d} \ mu (t)}$.

Finally, the modular function occurs in the definition of the involution on convolutional algebra${\ displaystyle L ^ {1} (G)}$ . In the space above, define for function${\ displaystyle L ^ {1}}$${\ displaystyle (G, \ mu)}$${\ displaystyle f, g \ in L ^ {1} (G)}$

${\ displaystyle f \ star g (t) \,: = \ int f (s) g (s ^ {- 1} t) \ mathrm {d} \ mu (s)}$

${\ displaystyle f ^ {*} (t): = \ Delta _ {G} (t ^ {- 1}) {\ overline {f (t ^ {- 1})}}}$.

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 . ${\ displaystyle f \ star g}$${\ displaystyle \ star}$${\ displaystyle f \ mapsto f ^ {*}}$${\ displaystyle L ^ {1} (G)}$

## Individual evidence

1. ^ Donald L. Cohn: Measure Theory. Birkhäuser, Boston MA 1980, ISBN 3-7643-3003-1 , set 9.3.4.
2. ^ Donald L. Cohn: Measure Theory. Birkhäuser, Boston MA 1980, ISBN 3-7643-3003-1 , chapter 9.3, exercise 2
3. ^ Donald L. Cohn: Measure Theory. Birkhäuser, Boston MA 1980, ISBN 3-7643-3003-1 , text after sentence 9.3.3.
4. Lynn H. Loomis : An Introduction to Abstract Harmonic Analysis. D. van Nostrand Co., Princeton NJ et al. 1953, § 30B.
5. ^ 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.