Modular function (harmonic analysis)

Unimodular Group is a redirect to this article. For groups of matrices with determinant 1 see special linear group .

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.