Banach Journal of Mathematical Analysis

Crossed products of algebras of unbounded operators

Maria Fragoulopoulou, Atsushi Inoue, and Klaus-Detlef Kursten

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Abstract

Consider a closed $O^*$--algebra $\mathcal{M}$ on a dense linear subspace $\mathcal{D}$ of a Hilbert space $\mathcal{H}$, a locally compact group $G$ with left invariant Haar measure $ds$ and an action $\alpha$ of $G$ on $\mathcal{M}$. Under some natural conditions, the $O^*$--crossed product $\mathcal{M}\underset{\alpha}{\overset{O^*}{\rtimes}}G$ of $\mathcal{M}$ and the $GW^*$--crossed product $\mathcal{M}\underset{\alpha}{\overset{GW^*}{\rtimes}}G$ are introduced. When $G$ is also abelian, the dual action $\widehat{\alpha}$ of the dual group $\widehat{G}$ on $\mathcal{M}\underset{\alpha}{\overset{O^*}{\rtimes}}G$ and on $\mathcal{M}\underset{\alpha}{\overset{GW^*}{\rtimes}}G$ is defined, which makes it possible to study the crossed products $(\mathcal{M}\underset{\alpha}{\overset{O^*}{\rtimes}}G)\underset{\widehat{\alpha}}{\overset{O^*}{\rtimes}}\widehat{G}$ and $\mathcal{M}\underset{\alpha}{\overset{GW^*}{\rtimes}}G)\underset{\widehat{\alpha}}{\overset{GW^*}{\rtimes}}\widehat{G}$. In case of modular actions, these constructions are used to obtain results on duality of type $\mathrm{II}$--like and type $\mathrm{III}$--like $GW^*$--algebras.

Article information

Source
Banach J. Math. Anal., Volume 9, Number 4 (2015), 316-358.

Dates
First available in Project Euclid: 17 April 2015

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1429286069

Digital Object Identifier
doi:10.15352/bjma/09-4-16

Mathematical Reviews number (MathSciNet)
MR3336895

Zentralblatt MATH identifier
1319.47068

Subjects
Primary: 47L65: Crossed product algebras (analytic crossed products)
Secondary: 47L60: Algebras of unbounded operators; partial algebras of operators 46L55: Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20] 46L60: Applications of selfadjoint operator algebras to physics [See also 46N50, 46N55, 47L90, 81T05, 82B10, 82C10]

Keywords
$O^*$--covariant system $O^*$--crossed product duality of $GW^*-$crossed products modular action

Citation

Fragoulopoulou, Maria; Inoue, Atsushi; Kursten, Klaus-Detlef. Crossed products of algebras of unbounded operators. Banach J. Math. Anal. 9 (2015), no. 4, 316--358. doi:10.15352/bjma/09-4-16. https://projecteuclid.org/euclid.bjma/1429286069


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