Twisted representations of product systems of $C^*$-correspondences: Wold decomposition and unitary extensions

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.

💡 Research Summary

This paper studies Wold‑type decompositions and unitary extension problems for multivariable isometric covariant representations associated with product systems of C∗‑correspondences. The authors begin by extending the classical von Neumann–Wold criterion, which characterizes when a single isometry splits into a unitary part and a shift part, to the setting of a tuple ((\sigma,T_{1},\dots,T_{n})) where each ((\sigma,T_{i})) is an isometric covariant representation of a C∗‑correspondence (E_{i}) over a fixed C∗‑algebra (A). The main result of Section 3 (Theorem 3.2) shows that such a tuple admits a unique Wold decomposition precisely when the induced (shift‑like) and fully co‑isometric (unitary‑like) components of each pair mutually reduce one another. This provides an operator‑theoretic necessary and sufficient condition that generalizes the von Neumann–Wold theorem to the categorical framework of product systems.

In Section 4 the authors introduce “twisted” and “doubly twisted” covariant representations. A twisted representation relaxes the usual commutation relations by inserting a prescribed family of unitaries ({U_{ij}}_{i\neq j}) so that \