Titlebook: An Invitation to Unbounded Representations of ?-Algebras on Hilbert Space; Konrad Schmüdgen Textbook 2020 The Editor(s) (if applicable) an

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-Algebras,n of real .-algebras, crossed product .-algebras, matrix algebras) and important classes (group graded .-algebras, hermitian and symmetric .-algebras) and examples (semigroup .-algebras, algebras defined by relations) of .-algebras are developed. Basic notions (positive linear functionals, states, c

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Integrable Representations of Commutative ,-Algebras,tions are characterized by a number of conditions. Each integrable representation of a finitely generated commutative unital .-algebra is expressed in terms of spectral integrals with respect to some spectral measure. An application to moment functionals is given.