Twisted representations of product systems of $C^*$-correspondences: Wold decomposition and unitary extensions
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| Format: | Preprint |
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2026
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| _version_ | 1866912823966171136 |
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| author | Solel, Baruch Suryawanshi, Mansi |
| author_facet | Solel, Baruch Suryawanshi, Mansi |
| contents | 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 $(σ, T_1, T_2, \ldots, T_n)$, where each $(σ,T_i)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_09391 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Twisted representations of product systems of $C^*$-correspondences: Wold decomposition and unitary extensions Solel, Baruch Suryawanshi, Mansi Operator Algebras Functional Analysis 46L08, 47A13, 47A20, 47A56, 47L05, 47L30, 47L55 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 $(σ, T_1, T_2, \ldots, T_n)$, where each $(σ,T_i)$ 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. |
| title | Twisted representations of product systems of $C^*$-correspondences: Wold decomposition and unitary extensions |
| topic | Operator Algebras Functional Analysis 46L08, 47A13, 47A20, 47A56, 47L05, 47L30, 47L55 |
| url | https://arxiv.org/abs/2601.09391 |