# Peter-Weyl theorem

A theorem on the approximation of functions on a compact topological group by means of representation functions (cf. Representation function). Let $\pi$ run through a family $\Sigma$ of representatives of all equivalence classes for the irreducible continuous unitary representations of a compact group $G$( cf. Representation of a topological group). Let $\mathop{\rm dim} \pi$ be the dimension of a representation $\pi$ and let $u _ {ij} ^ {( \pi ) }$ be its matrix elements in some orthonormal basis. The Peter–Weyl theorem asserts that the functions

$$\sqrt { \mathop{\rm dim} \pi } u _ {ij} ^ {( \pi ) } \ ( \pi \in \Sigma )$$

form an orthonormal basis in the space $L _ {2} (G)$ of square-summable functions with respect to the Haar measure on $G$( the measure of the entire group is taken to be 1). The algebra of all complex-valued representation functions on $G$, which coincides with the set of finite linear combinations of the functions $u _ {ij} ^ {( \pi ) }$, $\pi \in \Sigma$, is uniformly dense in the space of all continuous complex-valued functions in $G$.

If $G=T$ is the rotation group for the plane, this assertion coincides with an elementary theorem on approximating periodic continuous functions by trigonometric polynomials.

A consequence of the Peter–Weyl theorem is that the set of linear combinations of characters of the irreducible representations of $G$ is dense in the algebra of all continuous functions on $G$, constant on classes of conjugate elements. Another consequence is that for any element $a \in G$, $a \neq e$, there is an irreducible continuous representation $\phi$ of $G$ such that $\phi (a) \neq e$; if, on the other hand, $G$ is a compact Lie group, then $G$ has a faithful linear representation.

The Peter–Weyl theorem implies also the following more general assertion [5], [6]. Suppose one is given a continuous linear representation $\phi$ of a compact group $G$ in a Fréchet space $E$. Then the subspace of representation elements of $E$ is dense in $E$. Here an element $v \in E$ is called a representation, or spherical or almost-invariant, element if the orbit $\phi (G)v$ generates a finite-dimensional subspace in $E$. This is applicable in particular to the case where $E$ is the space of sections of a certain smoothness class of smooth vector $G$- fibrations, for example, the space of tensor fields of a certain type or given smoothness class on a smooth manifold with a smooth action of a compact Lie group $G$.

The Peter–Weyl theorem was proved in 1927 by F. Peter and H. Weyl .

#### References

 [1a] F. Peter, H. Weyl, "Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe" Math. Ann. , 97 (1927) pp. 737–755 [1b] F. Peter, H. Weyl, "Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe" , Gesammelte Abhandlungen H. Weyl , III : 73 , Springer (1968) pp. 58–75 [2] L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian) [3] E. Hewitt, K.A. Ross, "Abstract harmonic analysis" , 1–2 , Springer (1979) [4] C. Chevalley, "Theory of Lie groups" , 1 , Princeton Univ. Press (1946) [5] R.S. Palais, T.E. Stewart, "The cohomology of differentiable transformation groups" Amer. J. Math. , 83 : 4 (1961) pp. 623–644 [6] G.D. Mostow, "Cohomology of topological groups and solvmanifolds" Ann. of Math. , 73 : 1 (1961) pp. 20–48

A representation element is now usually called a $G$- finite element.
The statement that the algebra of complex-valued representation functions is uniformly dense in the algebra of continuous functions on $G$ is known as the Weyl approximation theorem. The Peter–Weyl theorem gives a complete description of the (left or right) regular representation in terms of its irreducible components. In particular, each irreducible component occurs with a multiplicity equal to its dimension, cf. [a1], Chapt. 7, §2. There exists a generalized Peter–Weyl theorem for unimodular Lie groups, cf. [a1], Chapt. 14, §2. The description of $L _ {2} (G)$( and the other unitary representations) in terms of the irreducible representations, including the fact that the irreducible unitary representations are finite dimensional, is known as Peter–Weyl theory, cf. e.g. [a2].