Peter-Weyl theorem

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A theorem on the approximation of functions on a compact topological group by means of representation functions (cf. Representation function). Let run through a family of representatives of all equivalence classes for the irreducible continuous unitary representations of a compact group (cf. Representation of a topological group). Let be the dimension of a representation and let be its matrix elements in some orthonormal basis. The Peter–Weyl theorem asserts that the functions

form an orthonormal basis in the space of square-summable functions with respect to the Haar measure on (the measure of the entire group is taken to be 1). The algebra of all complex-valued representation functions on , which coincides with the set of finite linear combinations of the functions , , is uniformly dense in the space of all continuous complex-valued functions in .

If 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 is dense in the algebra of all continuous functions on , constant on classes of conjugate elements. Another consequence is that for any element , , there is an irreducible continuous representation of such that ; if, on the other hand, is a compact Lie group, then 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 of a compact group in a Fréchet space . Then the subspace of representation elements of is dense in . Here an element is called a representation, or spherical or almost-invariant, element if the orbit generates a finite-dimensional subspace in . This is applicable in particular to the case where is the space of sections of a certain smoothness class of smooth vector -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 .

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