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Dept. - Trento University), Valter Moretti","submitted_at":"2012-05-21T07:32:55Z","abstract_excerpt":"Consider a finite dimensional complex Hilbert space $\\cH$, with $dim(\\cH) \\geq 3$, define $\\bS(\\cH):= \\{x\\in \\cH \\:|\\: ||x||=1\\}$, and let $\\nu_\\cH$ be the unique regular Borel positive measure invariant under the action of the unitary operators in $\\cH$, with $\\nu_\\cH(\\bS(\\cH))=1$. We prove that if a complex frame function $f : \\bS(\\cH)\\to \\bC$ satisfies $f \\in \\cL^2(\\bS(\\cH), \\nu_\\cH)$, then it verifies Gleason's statement: There is a unique linear operator $A: \\cH \\to \\cH$ such that $f(u) = < u| A u>$ for every $u \\in \\bS(\\cH)$. $A$ is Hermitean when $f$ is real. 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