Pontryagin duality

The Pontryagin duality , named after Lev Semjonowitsch Pontryagin , is a mathematical term from harmonic analysis . A locally compact Abelian group is assigned a further locally compact Abelian group as a dual group , so that the dual group for the dual group is again the starting group. This construction plays an important role in the abstract Fourier transform and the structural theory of the locally compact Abelian groups.

Pontryagin duality

The circle is with the multiplication as a group linking a compact group. ${\displaystyle {\mathbb {T} }=\{z\in {\mathbb {C} };|z|=1\}}$

If G is a locally compact Abelian group, a continuous group homomorphism is called a character of G. The dual group of G is the set of all characters of G. With multiplication it becomes an Abelian group, and the topology of compact convergence makes a locally compact group , d. H. to a topological group whose topology is locally compact. ${\displaystyle \chi :G\rightarrow {\mathbb {T} }}$ ${\displaystyle {\hat {G}}}$ ${\displaystyle (\chi \cdot \psi )(a):=\chi (a)\psi (a)}$ ${\displaystyle {\hat {G}}}$ ${\displaystyle {\hat {G}}}$

If there is a continuous homomorphism , then there is also a continuous homomorphism, the homomorphism that is too dual . ${\displaystyle \varphi \colon G\rightarrow H}$ ${\displaystyle {\hat {\varphi }}:{\hat {H}}\rightarrow {\hat {G}},\,{\hat {\varphi }}(\chi ):=\chi \circ \varphi }$ ${\displaystyle \varphi }$

Examples

• The characters of the remainder class group have the form , where . It applies if , and with it . ${\displaystyle {\mathbb {Z} }/n{\mathbb {Z} }}$ ${\displaystyle \chi _{m}:{\mathbb {Z} }/n{\mathbb {Z} }\rightarrow {\mathbb {T} },\,\chi _{m}([k])=e^{2\pi ikm/n}}$ ${\displaystyle m\in {\mathbb {Z} }}$ ${\displaystyle \chi _{m_{1}}=\chi _{m_{2}}}$ ${\displaystyle m_{1}+n{\mathbb {Z} }=m_{2}+n{\mathbb {Z} }}$ ${\displaystyle {\widehat {{\mathbb {Z} }/n{\mathbb {Z} }}}\cong {\mathbb {Z} }/n{\mathbb {Z} }}$
• Each character of has the shape for one . If one identifies with n, then is . ${\displaystyle {\mathbb {T} }}$ ${\displaystyle \chi _{n}(z)=z^{n}}$ ${\displaystyle n\in {\mathbb {Z} }}$ ${\displaystyle \chi _{n}}$ ${\displaystyle {\hat {\mathbb {T} }}\cong {\mathbb {Z} }}$
• The group has the characters , wherein . The assignment delivers . ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle \chi _{z}:{\mathbb {Z} }\rightarrow {\mathbb {T} }}$ ${\displaystyle \chi _{z}(n)=z^{n}}$ ${\displaystyle z\in {\mathbb {T} }}$ ${\displaystyle \chi _{z}\mapsto z}$ ${\displaystyle {\hat {\mathbb {Z} }}\cong {\mathbb {T} }}$
• ${\displaystyle \mathbb {R} }$with addition as a link and the Euclidean topology is a locally compact Abelian group. Each character has the shape for one . If one identifies with z, one has first as sets. This applies to all and the mapping is a homeomorphism , so one also has locally compact Abelian groups. ${\displaystyle \chi :{\mathbb {R} }\rightarrow {\mathbb {T} }}$ ${\displaystyle \chi _{z}(x)=e^{ixz}}$ ${\displaystyle z\in {\mathbb {R} }}$ ${\displaystyle \chi _{z}}$ ${\displaystyle {\hat {\mathbb {R} }}\cong {\mathbb {R} }}$ ${\displaystyle (\chi _{z}\cdot \chi _{w})(x)=\chi _{z+w}(x)}$ ${\displaystyle x\in {\mathbb {R} }}$ ${\displaystyle z\mapsto \chi _{z}}$ ${\displaystyle {\mathbb {R} }\cong {\hat {\mathbb {R} }}}$

Products by groups

If G and H are locally compact Abelian groups, so are their Cartesian products . Then define a character when you bet. In this way a group homeomorphism is obtained . ${\displaystyle G\times H}$ ${\displaystyle (\chi ,\psi )\in {\hat {G}}\times {\hat {H}}}$ ${\displaystyle G\times H}$ ${\displaystyle (\chi ,\psi )(x,y)=\chi (x)\cdot \psi (y)}$ ${\displaystyle {\widehat {G\times H}}\cong {\hat {G}}\times {\hat {H}}}$

This gives you many more examples:

• ${\displaystyle {\hat {G}}\cong G}$for every finite Abelian group G, because such is a finite product of groups of the form (see: Finite Abelian Group ). ${\displaystyle {\mathbb {Z} }/n{\mathbb {Z} }}$
• ${\displaystyle {\widehat {{\mathbb {Z} }^{n}}}\cong {\mathbb {T} }^{n}}$, , ${\displaystyle {\widehat {{\mathbb {T} }^{n}}}\cong {\mathbb {Z} }^{n}}$ ${\displaystyle {\widehat {{\mathbb {R} }^{n}}}\cong {\mathbb {R} }^{n}}$

Pontryagin's duality theorem

You have a natural image . The set of Pontrjagin states that this figure is always a topological group isomorphism is. This justifies the designation of the dual group of G, because according to the above sentence, G can be recovered from by forming a new dual group. ${\displaystyle \Phi :G\rightarrow {\hat {\hat {G}}},\,(\Phi (x))(\chi ):=\chi (x)}$ ${\displaystyle {\hat {G}}}$

Relationships between group and dual group

Due to the Pontryagin duality, one expects a series of relationships between a locally compact Abelian group G and its dual group . Thereby one finds relationships between algebraic and topological properties. The following applies as an example: ${\displaystyle {\hat {G}}}$

• G is discrete is compact . ${\displaystyle \Leftrightarrow }$ ${\displaystyle {\hat {G}}}$
• G is compact is discrete . ${\displaystyle \Leftrightarrow }$ ${\displaystyle {\hat {G}}}$

For a compact group, the following statements are equivalent:

• G is connected .
• G is divisible .
• ${\displaystyle {\hat {G}}}$is torsion-free .

Another related property leads to the following equivalence:

• A compact group G is totally disconnected if and only if is a divisible group . ${\displaystyle {\hat {G}}}$

A continuous homomorphism is called strict if the mapping is open , i.e. H. the picture of any open set is relatively open in the picture of . A sequence of homomorphisms is called strict if every homomorphism is strict. Finally, if one denotes the one-element group with 1 and observes , then the following theorem applies: ${\displaystyle \varphi :G\rightarrow H}$ ${\displaystyle \varphi }$ ${\displaystyle G\rightarrow \varphi (G)}$ ${\displaystyle \varphi }$ ${\displaystyle \ldots \rightarrow G\rightarrow H\rightarrow \ldots }$ ${\displaystyle {\hat {1}}\cong 1}$

• Let be a sequence of continuous homomorphisms between locally compact Abelian groups. Then the following statements are equivalent: ${\displaystyle 1\rightarrow U\rightarrow G\rightarrow H\rightarrow 1}$
  • ${\displaystyle 1\rightarrow U\rightarrow G\rightarrow H\rightarrow 1}$is a strict and exact sequence .
  • ${\displaystyle 1\rightarrow {\hat {H}}\rightarrow {\hat {G}}\rightarrow {\hat {U}}\rightarrow 1}$is a strict and exact sequence .

From this one draws further conclusions:

• A continuous homomorphism is strict if and only if is strict. ${\displaystyle \varphi :G\rightarrow H}$ ${\displaystyle {\hat {\varphi }}:{\hat {H}}\rightarrow {\hat {G}}}$
• If there is a closed subgroup, then is . Here is the dual mapping for inclusion . ${\displaystyle U\subset G}$ ${\displaystyle {\widehat {G/U}}\cong \ker({\hat {G}}\rightarrow {\hat {U}})=\{\chi \in {\hat {G}};\chi |_{U}=1\}}$ ${\displaystyle {\hat {G}}\rightarrow {\hat {U}}}$ ${\displaystyle U\subset G}$

Compact generated groups

The Pontryagin duality is an important tool in structural theory for locally compact Abelian groups. A locally compact group is said to be compact if there is a compact subset of G that generates G as a group . A discrete group is created compact if and only if it is finitely created.

For a locally compact Abelian group are equivalent:

• G is generated compact.
• ${\displaystyle G\cong {\mathbb {R} }^{m}\times {\mathbb {Z} }^{n}\times K}$, where and K is a compact group. ${\displaystyle m,n\in {\mathbb {N} }_{0}}$
• ${\displaystyle {\hat {G}}\cong {\mathbb {R} }^{m}\times {\mathbb {T} }^{n}\times D}$, where and D is a discrete group. ${\displaystyle m,n\in {\mathbb {N} }_{0}}$

Addition: The numbers m and n are uniquely determined by G and K is the largest compact subgroup of G.

Gelfand transformation

As explained in the article Harmonic Analysis , the dual group of a locally compact Abelian group G occurs in the Gelfand transform of the convolutional algebra over G.

Pontryagin duality as functor

The Pontryagin duality, i.e. H. the above-described assignments and of locally compact Abelian groups and continuous homomorphisms is obviously a contravariant functor . Executing this functor twice in a row leads to the identical functor (more precisely: to a natural equivalence to the identical functor). ${\displaystyle G\mapsto {\hat {G}}}$ ${\displaystyle \varphi \mapsto {\hat {\varphi }}}$

literature

• Lynn H. Loomis : An Introduction to Abstract Harmonic Analysis , D. van Nostrand Co, 1953
• Walter Rudin : Fourier Analysis on Groups , 1962
• E. Hewitt, K. Ross: Abstract Harmonic Analysis I, II , Springer (1963), (1970).