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 Ferdinand Georg Frobenius and Issai Schur. Let G be a compact group. 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). In the case that G is the group of unit complex numbers, this last result is simply a standard result from Fourier series.
Matrix coefficients A matrix coefficient of the group G is a complex-valued function φ {\displaystyle \varphi } on G given as the composition
φ = L ∘ π {\displaystyle \varphi =L\circ \pi }
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. The first part of the Peter–Weyl theorem asserts (Bump 2004, §4.1; Knapp 1986, Theorem 1.12):
Peter–Weyl Theorem (Part I). The set of matrix coefficients of G is dense in the space of continuous complex functions C(G) on G, equipped with the uniform norm. This first result resembles the Stone–Weierstrass theorem in that it indicates the density of a set of functions in the space of all continuous functions, subject only to an algebraic characterization. In fact, the matrix coefficients form a unital algebra invariant under complex conjugation because the product of two matrix coefficients is a matrix coefficient of the tensor product representation, and the complex conjugate is a matrix coefficient of the dual representation. Hence the theorem follows directly from the Stone–Weierstrass theorem if the matrix coefficients separate points, which is obvious if G is a matrix group (Knapp 1986, p. 17). Conversely, it is a consequence of the theorem that any compact Lie group is isomorphic to a matrix group (Knapp 1986, Theorem 1.15). A corollary of this result is that the matrix coefficients of G are dense in L2(G).
Decomposition of a unitary representation The second part of the theorem gives the existence of a decomposition of a unitary representation of G into finite-dimensional representations. Now, intuitively groups were conceived as rotations on geometric objects, so it is only natural to study representations which essentially arise from continuous actions on Hilbert spaces. (For those who were first introduced to dual groups consisting of characters which are the continuous homomorphisms into the circle group, this approach is similar except that the circle group is (ultimately) generalised to the group of unitary operators on a given Hilbert space.) Let G be a topological group and H a complex Hilbert space. A continuous linear action ∗ : G × H → H, gives rise to a continuous map ρ∗ : G → HH (functions from H to H with the strong topology) defined by: ρ∗(g)(v) = ∗(g,v). This map is clearly a homomorphism from G into GL(H), the bounded linear operators on H. Conversely, given such a map, we can uniquely recover the action in the obvious way. Thus we define the representations of G on a Hilbert space H to be those group homomorphisms, ρ, which arise from continuous actions of G on H. We say that a representation ρ is unitary if ρ(g) is a unitary operator for all g ∈ G; i.e., ⟨ ρ ( g ) v , ρ ( g ) w ⟩ = ⟨ v , w ⟩ {\displaystyle \langle \rho (g)v,\rho (g)w\rangle =\langle v,w\rangle } for all v, w ∈ H. (I.e. it is unitary if ρ : G → U(H). Notice how this generalises the special case of the one-dimensional Hilbert space, where U(C) is just the circle group.) Given these definitions, we can state the second part of the Peter–Weyl theorem (Knapp 1986, Theorem 1.12):
Peter–Weyl Theorem (Part II). Let ρ be a unitary representation of a compact group G on a complex Hilbert space H. Then H splits into an orthogonal direct sum of irreducible finite-dimensional unitary representations of G.
Decomposition of square-integrable functions To state the third and final part of the theorem, there is a natural Hilbert space over G consisting of square-integrable functions, L 2 ( G ) {\displaystyle L^{2}(G)} ; this makes sense because the Haar measure exists on G. The group G has a unitary representation ρ on L 2 ( G ) {\displaystyle L^{2}(G)} given by acting on the left, via
ρ ( h ) f ( g ) = f ( h − 1 g ) . {\displaystyle \rho (h)f(g)=f(h^{-1}g).}
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