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<p>We introduce operator-valued twisted Araki–Woods algebras. These are operator-valued versions of a class of second quantization algebras that includes <i>q</i>-Gaussian and <i>q</i>-Araki–Woods algebras and also generalize Shlyakhtenko’s von Neumann algebras generated by operator-valued semicircular variables. We develop a disintegration theory that reduces the isomorphism type of operator-valued twisted Araki–Woods algebras over type <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5285_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="8" /> </InlineMediaObject> <EquationSource Format="TEX">\(\textrm{I}\)</EquationSource> <EquationSource Format="MATHML"><math> <mtext>I</mtext> </math></EquationSource> </InlineEquation> factors to the scalar-valued case. Moreover, these algebras come with a natural weight, and we characterize its modular theory. We also give sufficient criteria that guarantee the factoriality of these algebras.</p>

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Operator-Valued Twisted Araki–Woods Algebras

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年4月10日

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  • R. Rahul Kumar,
  • Melchior Wirth

摘要

We introduce operator-valued twisted Araki–Woods algebras. These are operator-valued versions of a class of second quantization algebras that includes q-Gaussian and q-Araki–Woods algebras and also generalize Shlyakhtenko’s von Neumann algebras generated by operator-valued semicircular variables. We develop a disintegration theory that reduces the isomorphism type of operator-valued twisted Araki–Woods algebras over type \(\textrm{I}\) I factors to the scalar-valued case. Moreover, these algebras come with a natural weight, and we characterize its modular theory. We also give sufficient criteria that guarantee the factoriality of these algebras.