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Compact non-abelian groups

The Fourier transform can also be defined for functions on a non-abelian group, provided that the group is compact. Removing the assumption that the underlying group is abelian, irreducible unitary representations need not always be one-dimensional. This means the Fourier transform on a non-abelian group takes values as Hilbert space operators. The Fourier transform on compact groups is a major tool in representation theory and non-commutative harmonic analysis.

Let G be a compact Hausdorff topological group. Let Σ denote the collection of all isomorphism classes of finite-dimensional irreducible unitary representations, along with a definite choice of representation U on the Hilbert space H of finite dimension d for each σ ∈ Σ. If μ is a finite Borel measure on G, then the Fourier–Stieltjes transform of μ is the operator on H defined by ⟨ μ ^ ξ, η ⟩ H σ = ∫ G ⟨ U ¯ g (σ) ξ, η ⟩ d μ (g) {\displaystyle \left\langle {\hat {\mu }}\xi,\eta \right\rangle _{H_{\sigma }}=\int _{G}\left\langle {\overline {U}}_{g}^{(\sigma)}\xi,\eta \right\rangle \,d\mu (g)} where U is the complex-conjugate representation of U acting on H. If μ is absolutely continuous with respect to the left-invariant probability measure λ on G, represented as d μ = f d λ {\displaystyle d\mu =f\,d\lambda } for some f ∈ L (λ), one identifies the Fourier transform of f with the Fourier–Stieltjes transform of μ.

The mapping μ ↦ μ ^ {\displaystyle \mu \mapsto {\hat {\mu }}} defines an isomorphism between the Banach space M (G) of finite Borel measures (see rca space) and a closed subspace of the Banach space C (Σ) consisting of all sequences E = (E) indexed by Σ of (bounded) linear operators E : H → H for which the norm ‖ E ‖ = sup σ ∈ Σ ‖ E σ ‖ {\displaystyle \|E\|=\sup _{\sigma \in \Sigma }\left\|E_{\sigma }\right\|} is finite. The " convolution theorem " asserts that, furthermore, this isomorphism of Banach spaces is in fact an isometric isomorphism of C*-algebras into a subspace of C (Σ). Multiplication on M (G) is given by convolution of measures and the involution * defined by f ∗ (g) = f (g − 1) ¯, {\displaystyle f^{*}(g)={\overline {f\left(g^{-1}\right)}},} and C (Σ) has a natural C * -algebra structure as Hilbert space operators.

The Peter–Weyl theorem holds, and a version of the Fourier inversion formula (Plancherel's theorem) follows: if f ∈ L (G), then f (g) = ∑ σ ∈ Σ d σ tr (f ^ (σ) U g (σ)) {\displaystyle f(g)=\sum _{\sigma \in \Sigma }d_{\sigma }\operatorname {tr} \left({\hat {f}}(\sigma)U_{g}^{(\sigma)}\right)} where the summation is understood as convergent in the L sense.

The generalization of the Fourier transform to the noncommutative situation has also in part contributed to the development of noncommutative geometry. In this context, a categorical generalization of the Fourier transform to noncommutative groups is Tannaka–Krein duality, which replaces the group of characters with the category of representations. However, this loses the connection with harmonic functions.