dorsal/arxiv
View SchemaTwisted representations of product systems of $C^*$-correspondences: Wold decomposition and unitary extensions
| Authors | Baruch Solel, Mansi Suryawanshi |
|---|---|
| Categories | |
| ArXiv ID | 2601.09391vv1 |
| URL | https://arxiv.org/abs/2601.09391 |
| License | http://creativecommons.org/licenses/by/4.0/ |
Abstract
We investigate Wold-type decompositions and unitary extension problems for multivariable isometric covariant representations associated with product systems of $C^*$-correspondences. First, we establish an operator-theoretic characterization for the existence of a Wold decomposition for the tuple $(\sigma, T_1, T_2, \ldots, T_n)$, where each $(\sigma,T_i)$ is an isometric covariant representation of a $C^*$\nobreakdash-correspondence. We then introduce twisted and doubly twisted covariant representations of product systems. For doubly twisted isometric representations, we prove the existence of a Wold decomposition, recovering earlier results for doubly commuting representations as special cases. We further obtain explicit descriptions of the resulting Wold summands and develop concrete Fock-type models realizing each component. We present non-trivial examples of these families. Finally, we construct unitary extensions via a direct-limit procedure. As applications, we obtain unitary extensions for several previously studied classes of operator tuples, including doubly twisted, doubly non-commuting, and doubly commuting isometries, and for a special class of doubly twisted representations of product system.
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"abstract": "We investigate Wold-type decompositions and unitary extension problems for multivariable isometric covariant representations associated with product systems of $C^*$-correspondences. First, we establish an operator-theoretic characterization for the existence of a Wold decomposition for the tuple $(\\sigma, T_1, T_2, \\ldots, T_n)$, where each $(\\sigma,T_i)$ is an isometric covariant representation of a $C^*$\\nobreakdash-correspondence. We then introduce twisted and doubly twisted covariant representations of product systems. For doubly twisted isometric representations, we prove the existence of a Wold decomposition, recovering earlier results for doubly commuting representations as special cases. We further obtain explicit descriptions of the resulting Wold summands and develop concrete Fock-type models realizing each component. We present non-trivial examples of these families. Finally, we construct unitary extensions via a direct-limit procedure. As applications, we obtain unitary extensions for several previously studied classes of operator tuples, including doubly twisted, doubly non-commuting, and doubly commuting isometries, and for a special class of doubly twisted representations of product system.",
"arxiv_id": "2601.09391",
"authors": [
"Baruch Solel",
"Mansi Suryawanshi"
],
"categories": [
"math.OA",
"math.FA"
],
"license": "http://creativecommons.org/licenses/by/4.0/",
"title": "Twisted representations of product systems of $C^*$-correspondences: Wold decomposition and unitary extensions",
"url": "https://arxiv.org/abs/2601.09391",
"version": "v1"
},
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"id": "arXiv Dataset",
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"variant": "snapshot-2026-01-17",
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