Peter-Weyl's theorem

In mathematical branch of harmonic analysis of generalized set of Peter-Weyl , named after Hermann Weyl and his student Fritz Peter (1899-1949), the Fourier series for functions on arbitrary compact topological groups .

Representations on compact groups

Be a compact topological group. For a complex Hilbert space, a continuous group homomorphism is called the representation of the group, with the weak operator topology being provided. It can now be shown that each such has a compact self-adjoint commutation operator and thus a finite-dimensional, nontrivial invariant subspace of as the eigenspace of this operator . Therefore every irreducible representation of a compact group is finite-dimensional and every representation can be represented as a direct sum of such, i.e. has a decomposition into irreducible representations . ${\displaystyle G}$ ${\displaystyle H_{\pi }}$ ${\displaystyle \pi \colon G\to U(H_{\pi })}$ ${\displaystyle U(H_{\pi })}$ ${\displaystyle \pi }$ ${\displaystyle H_{\pi }}$

The left-regular representation is of particular interest ; this is defined by, where and is a function and is a square-integrable function and with respect to the left-invariant to normalized hair measurement . It can be shown that for each such function the function given by the above formula can again be square-integrable and that two functions that are almost everywhere the same are mapped to almost everywhere the same functions, i.e. overall it actually determines an operator whose unitarity can easily be demonstrated. Similarly, the right-regular display is defined by and the two-sided display is defined by. ${\displaystyle \mathbf {L} \colon G\to U(L^{2}(G))}$ ${\displaystyle (\mathbf {L} (g)f)(x)=f(g^{-1}x)}$ ${\displaystyle g,x\in G}$ ${\displaystyle f\in L^{2}(G)}$ ${\displaystyle 1}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle \mathbf {L} (g)f\colon G\rightarrow \mathbb {C} }$ ${\displaystyle \mathbf {L} (g)}$ ${\displaystyle L^{2}(G)}$ ${\displaystyle (\mathbf {R} (g)f)(x)=f(xg)}$ ${\displaystyle \mathbf {\mathbf {L} \times \mathbf {R} } \colon G\times G\to U(L^{2}(G)),\;((\mathbf {L} \times \mathbf {R} )(g,h)f)(x)=f(g^{-1}xh)}$

For every representation and is a bounded continuous function called the matrix coefficient ( see Fourier-Stieltjes algebra ). ${\displaystyle \pi }$ ${\displaystyle u,v\in H_{\pi }}$ ${\displaystyle \pi _{uv}\colon G\to \mathbb {C} ,\;g\mapsto \langle \pi (g)v,u\rangle }$

Fourier transform

From all irreducible representations of, choose a system of representatives with respect to unitary equivalence . Each representation corresponds to a Hilbert space representation of the Banach - * - algebra with convolution (the so-called group algebra ), so that the equation ${\displaystyle G}$ ${\displaystyle {\hat {G}}}$ ${\displaystyle \pi \colon G\to U(L^{2}(G))}$ ${\displaystyle \pi \colon L^{1}(G)\to L(L^{2}(G))}$ ${\displaystyle L^{1}(G)}$ ${\displaystyle u,v\in H_{\pi }}$

${\displaystyle \langle \pi (f)u,v\rangle =\int _{G}\langle \pi (g)u,v\rangle f(g)\mathrm {d} g}$

consists. Since the hair measure is finite on a compact group is . The Fourier transform for a function is now defined as , where is a mapping of into the orthogonal sum ${\displaystyle L^{2}(G)\subseteq L^{1}(G)}$ ${\displaystyle f\in L^{2}(G)}$ ${\displaystyle {\mathcal {F}}(f)=\left(\pi (f)\right)_{\pi \in {\hat {G}}}}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle L^{2}(G)}$

${\displaystyle H:=\bigoplus _{\pi \in {\hat {G}}}HS(H_{\pi })}$

of the spaces of matrices , equipped with the Hilbert-Schmidt scalar product (this is always possible in the compact case, since the representation spaces are finite-dimensional). ${\displaystyle H_{\pi }}$

sentence

Peter-Weyl's theorem now states that the Fourier transform of a compact group is unitary up to certain constant factors , and constructs the inverse mapping . Is more accurate

${\displaystyle {\mathcal {F}}^{\prime }\colon L^{2}(G)\to H,\quad f\mapsto \left({\sqrt {\dim H_{\pi }}}{\mathcal {F}}(f)_{\pi }\right)_{\pi \in {\hat {G}}}}$

unitary. The reverse mapping is given by

${\displaystyle {\mathcal {F}}^{\prime *}\colon H\to L^{2}(G),\quad (T_{\pi })_{\pi \in {\hat {G}}}\mapsto \left(f\colon x\mapsto \sum _{\pi \in {\hat {G}}}{\sqrt {\dim H_{\pi }}}\operatorname {Tr} (T_{\pi }\pi (x)^{*})\right)}$,

where the trace denotes and the sum is to be understood in the sense of unconditional convergence . ${\displaystyle \operatorname {Tr} }$

Partial statements

Here some partial statements are given, which are sometimes used for proof, and which in turn follow directly from Peter-Weyl's theorem in the above form.

The spaces are pairwise orthogonal subspaces of , so the subspaces are also pairwise orthogonal and the operator is also unitary. If the family is an orthonormal basis of , then the family of all dyadic products is an orthonormal basis of and thus an orthonormal basis of . If, accordingly, orthonormal bases are given for each , the functions form an orthonormal basis of . ${\displaystyle HS(\pi )}$ ${\displaystyle H}$ ${\displaystyle L_{\pi }^{2}:={\mathcal {F}}^{\prime *}(HS(\pi ))\subseteq L^{2}(G)}$ ${\displaystyle U_{\pi }:={\mathcal {F}}^{\prime *}|_{HS(\pi )\to L_{\pi }^{2}}}$ ${\displaystyle (e_{i})}$ ${\displaystyle H_{\pi }}$ ${\displaystyle (e_{i}\otimes e_{j})_{i,j}}$ ${\displaystyle HS(H_{\pi })}$ ${\displaystyle (U_{\pi }(e_{i}\otimes e_{j}))_{i,j}=({\sqrt {\dim H_{\pi }}}\pi _{e_{i}e_{j}})_{i,j}}$ ${\displaystyle L_{\pi }^{2}}$ ${\displaystyle (e_{i}^{\pi })_{i}}$ ${\displaystyle \pi \in {\hat {G}}}$ ${\displaystyle ({\sqrt {\dim H_{\pi }}}\pi _{e_{i}^{\pi }e_{j}^{\pi }})_{\pi ,i,j}}$ ${\displaystyle L^{2}(G)}$

The representation is defined as the outer tensor with the contragredient representation , concretely: ${\displaystyle \pi ^{2}\colon G\times G\to HS(H_{\pi })}$ ${\displaystyle \pi ^{2}:=\pi \otimes {\bar {\pi }}}$

${\displaystyle \pi ^{2}(g,h)T=\pi (g)T\pi (h)^{*}}$.

The operator is now a commutation operator between and , i.e. H. ${\displaystyle U_{\pi }}$ ${\displaystyle \pi ^{2}}$ ${\displaystyle \mathbf {L} \times \mathbf {R} }$

${\displaystyle (\mathbf {L} \times \mathbf {R} )(g,h)U_{\pi }=U_{\pi }\pi ^{2}(g,h)}$,

which is equivalent to the two-sided representation restricted to . If one chooses fixed and normalized , then the image of the operator is ${\displaystyle \pi ^{2}}$ ${\displaystyle L_{\pi }^{2}}$ ${\displaystyle u\in H_{\pi }}$

${\displaystyle U_{\pi }^{u}:=U|_{\left\{v\otimes u\mid v\in H_{\pi }\right\}}}$

invariant under the left-regular representation, the unitary operator (if the image space is restricted)

${\displaystyle V_{\pi }^{u}:=U_{\pi }^{u}(v\mapsto v\otimes u)}$

is a Vertauschungsoperator between and , . Thus, every irreducible representation of a compact group is equivalent to a partial representation of the left-regular representation. The multiplicity of the representation in the left-regular representation, that is, how often it occurs in a decomposition of this into irreducible, is exactly equal to the dimension of the representation space . The orthogonal projection is given by a fold . These results apply in a completely analogous manner to the right-regular representation by considering the reverse convolution instead of and during the projection. ${\displaystyle \pi }$ ${\displaystyle \mathbf {L} }$ ${\displaystyle \mathbf {L} (x)V_{\pi }^{u}=V_{\pi }^{u}\pi (x)}$ ${\displaystyle \pi }$ ${\displaystyle \dim H_{\pi }}$ ${\displaystyle p\colon L^{2}(G)\to \operatorname {Im} (V_{\pi }^{u})=\operatorname {Im} (U_{\pi }^{u})}$ ${\displaystyle f\mapsto f*U_{\pi }^{u}u={\sqrt {\dim H_{\pi }}}f*\pi _{uu}}$ ${\displaystyle u\otimes v}$ ${\displaystyle v\otimes u}$

example

Be the circle group . Since it is Abelian , every irreducible representation is a character , that is, a mapping into the circle group itself. These are given by the functions for . For and applies ${\displaystyle G=\mathbb {S} =U(1)}$ ${\displaystyle \mathbb {S} }$ ${\displaystyle \chi _{m}\colon x\mapsto x^{m}}$ ${\displaystyle m\in \mathbb {Z} }$ ${\displaystyle f\in L^{2}(\mathbb {S} )}$ ${\displaystyle u,v\in \mathbb {C} }$

${\displaystyle \chi _{m}(f)u{\bar {v}}=\langle \chi _{m}(f)u,v\rangle =\int _{\mathbb {S} }\langle \chi _{m}(x)u,v\rangle f(x)\mathrm {d} x=\int _{\mathbb {S} }\chi _{m}(x)u{\bar {v}}f(x)\mathrm {d} x}$

and therefore easy . This is nothing more than the well-known -th Fourier coefficient to . Peter-Weyl's theorem provides (since the representation space is one-dimensional, no further scaling is necessary) the unitarity of this transformation in space as well as the inversion ${\displaystyle \chi _{m}(f)=\int _{\mathbb {S} }\chi _{m}(x)f(x)\mathrm {d} x}$ ${\displaystyle m}$ ${\displaystyle f}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \bigoplus _{\mathbb {Z} }HS(\mathbb {C} )\cong \ell ^{2}(\mathbb {Z} )}$

${\displaystyle (c_{m})_{m\in \mathbb {Z} }\mapsto \left(x\mapsto \sum _{m\in \mathbb {Z} }\operatorname {Tr} (\chi _{m}(f)\chi _{m}(x)^{*})=\sum _{m\in \mathbb {Z} }\chi _{m}(f)x^{-m}\right)}$.

literature

• Gerald Budge Folland: A Course in Abstract Harmonic Analysis . CRC Press , 1995, ISBN 0-8493-8490-7 , pp. 128 ff .
• Anton Deitmar, Siegfried Echterhoff: Principles of Harmonic Analysis . Springer , 2009, ISBN 978-0-387-85468-7 , pp. 141 ff ., Doi : 10.1007 / 978-0-387-85469-4 .
• Mitsuo Sugiura: Unit Representations and Harmonic Analysis . 2nd Edition. North-Holland, 1990, ISBN 0-444-88593-5 , pp. 19 ff .
• F. Peter, H. Weyl: About the completeness of the primitive representations of a closed continuous group. Mathematische Annalen, Volume 97, 1927, pp. 737-755. (on-line)