Theorem. Spectral bridge for split abelian extensions [connes-0004]

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spectral_criterion

L∃∀NAGENTDRAFTED

A split extension \(A\rtimes H\) with \(A\) abelian turns the restriction of a unitary representation to \(A\) into harmonic analysis on the Pontryagin dual \(\widehat A\). For a vector \(\xi \), a projection-valued spectral measure \(P\) yields the positive scalar measure \[\mu _\xi (B)=\langle P(B)\xi ,\xi \rangle .\] Its total mass is \(\lVert \xi \rVert ^2\); kernel displacement becomes the energy \(\int _{\widehat A}|\chi (a)-1|^2\,d\mu _\xi (\chi )\); and quotient-fixed vectors give invariant measures. A positive atom at the trivial character yields a nonzero invariant vector through its spectral projection.