Peter–Weyl theorem

In mathematics, the Peter–Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian. It was initially proved by Hermann Weyl, with his student Fritz Peter, in the setting of a compact topological group G (Peter & Weyl 1927). The theorem is a collection of results generalizing the significant facts about the decomposition of the regular representation of any finite group, as discovered by F. G. Frobenius and Issai Schur.

The theorem has three parts. The first part states that the matrix coefficients of irreducible representations of G are dense in the space C(G) of continuous complex-valued functions on G, and thus also in the space L²(G) of square-integrable functions. The second part asserts the complete reducibility of unitary representations of G. The third part then asserts that the regular representation of G on L²(G) decomposes as the direct sum of all irreducible unitary representations. Moreover, the matrix coefficients of the irreducible unitary representations form an orthonormal basis of L²(G).

A matrix coefficient of the group G is a complex-valued function φ on G given as the composition

where π : G → GL(V) is a finite-dimensional (continuous) group representation of G, and L is a linear functional on the vector space of endomorphisms of V (e.g. trace), which contains GL(V) as an open subset. Matrix coefficients are continuous, since representations are by definition continuous, and linear functionals on finite-dimensional spaces are also continuous.

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