, Volume 14, Issue 5, pp 1435-1443
Date: 17 Nov 2012

Generalized Complex Spherical Harmonics, Frame Functions, and Gleason Theorem

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Consider a finite dimensional complex Hilbert space \({\mathcal{H}}\) , with \({dim(\mathcal{H}) \geq 3}\) , define \({\mathbb{S}(\mathcal{H}):= \{x\in \mathcal{H} \:|\: \|x\|=1\}}\) , and let \({\nu_\mathcal{H}}\) be the unique regular Borel positive measure invariant under the action of the unitary operators in \({\mathcal{H}}\) , with \({\nu_\mathcal{H}(\mathbb{S}(\mathcal{H}))=1}\) . We prove that if a complex frame function \({f : \mathbb{S}(\mathcal{H})\to \mathbb{C}}\) satisfies \({f \in \mathbb{L}^2(\mathbb{S}(\mathcal{H}), \nu_\mathcal{H})}\) , then it verifies Gleason’s statement: there is a unique linear operator \({A: \mathcal{H} \to \mathcal{H}}\) such that \({f(u) = \langle u| A u\rangle}\) for every \({u \in \mathbb{S}(\mathcal{H}).\,A}\) is Hermitean when f is real. No boundedness requirement is thus assumed on f a priori.

Communicated by Klaus Fredenhagen.