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<p>We show how the Gelfand spectrum of certain commutative operator algebras can be studied based on the theorem of Stone and von Neumann. The method presented is a natural addition to the tools of <i>quantum spectral synthesis</i>, which were recently used to characterize certain commutative Toeplitz algebras on the Fock space. Our method applies to this setting and also to more general abelian phase spaces. Besides characterizing Gelfand spectra of such commutative operator algebras, we also prove an extension of this result to the operator-valued case.</p>

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Commutative C\(^*\) algebras and Gelfand theory through phase space methods

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  • 出版模式：

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  • 发表日期：

    2025年9月18日

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  • Robert Fulsche,
  • Oliver Fürst

摘要

We show how the Gelfand spectrum of certain commutative operator algebras can be studied based on the theorem of Stone and von Neumann. The method presented is a natural addition to the tools of quantum spectral synthesis, which were recently used to characterize certain commutative Toeplitz algebras on the Fock space. Our method applies to this setting and also to more general abelian phase spaces. Besides characterizing Gelfand spectra of such commutative operator algebras, we also prove an extension of this result to the operator-valued case.