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Stone–von Neumann theorem
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=== Reformulation via Fourier transform === Let {{mvar|G}} be a [[locally compact abelian group]] and {{math|''G''<sup>^</sup>}} be the [[Pontryagin dual]] of {{mvar|G}}. The [[Fourier transform|Fourier–Plancherel transform]] defined by <math display="block">f \mapsto {\hat f}(\gamma) = \int_G \overline{\gamma(t)} f(t) d \mu (t)</math> extends to a C*-isomorphism from the [[group algebra of a locally compact group|group C*-algebra]] {{math|C*(''G'')}} of {{mvar|G}} and {{math|C<sub>0</sub>(''G''<sup>^</sup>)}}, i.e. the [[Spectrum of a C*-algebra|spectrum]] of {{math|C*(''G'')}} is precisely {{math|''G''<sup>^</sup>}}. When {{mvar|G}} is the real line {{math|'''R'''}}, this is Stone's theorem characterizing one-parameter unitary groups. The theorem of Stone–von Neumann can also be restated using similar language. The group {{mvar|G}} acts on the {{mvar|C}}*-algebra {{math|C<sub>0</sub>(''G'')}} by right translation {{mvar|ρ}}: for {{mvar|s}} in {{mvar|G}} and {{mvar|f}} in {{math|C<sub>0</sub>(''G'')}}, <math display="block">(s \cdot f)(t) = f(t + s).</math> Under the isomorphism given above, this action becomes the natural action of {{mvar|G}} on {{math|C*(''G''<sup>^</sup>)}}: <math display="block"> \widehat{ (s \cdot f) }(\gamma) = \gamma(s) \hat{f} (\gamma).</math> So a covariant representation corresponding to the {{mvar|C}}*-[[crossed product]] <math display="block">C^*\left( \hat{G} \right) \rtimes_{\hat{\rho}} G </math> is a unitary representation {{math|''U''(''s'')}} of {{mvar|G}} and {{math|''V''(''γ'')}} of {{math|''G''<sup>^</sup>}} such that <math display="block">U(s) V(\gamma) U^*(s) = \gamma(s) V(\gamma).</math> It is a general fact that covariant representations are in one-to-one correspondence with *-representation of the corresponding crossed product. On the other hand, all [[irreducible representation]]s of <math display="block">C_0(G) \rtimes_\rho G </math> are unitarily equivalent to the <math>{\mathcal K}\left(L^2(G)\right)</math>, the [[compact operator on Hilbert space|compact operators]] on {{math|''L''<sup>2</sup>(''G''))}}. Therefore, all pairs {{math|{''U''(''s''), ''V''(''γ'')} }} are unitarily equivalent. Specializing to the case where {{math|1=''G'' = '''R'''}} yields the Stone–von Neumann theorem.
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