Orthogonal decompositions and twisted isometries (2104.07628v3)

Abstract: Let $n > 1$. Let ${U_{ij}}{1 \leq i < j \leq n}$ be $\binom{n}{2}$ commuting unitaries on some Hilbert space $\mathcal{H}$, and suppose $U{ji} := U_{ij}^*$, $1 \leq i < j \leq n$. An $n$-tuple of isometries $V = (V_1, \ldots ,V_n)$ on $\mathcal{H}$ is called $\mathcal{U}n$-twisted isometry with respect to ${U{ij}}{i<j}$ (or simply $\mathcal{U}_n$-twisted isometry if ${U{ij}}{i<j}$ is clear from the context) if $V_i$'s are in the commutator ${U{st}: s \neq t}'$, and $V_i^V_j=U_{ij}^*V_jV_i^$, $i \neq j$ We prove that each $\mathcal{U}_n$-twisted isometry admits a von Neumann-Wold type orthogonal decomposition, and prove that the universal $C^*$-algebra generated by $\mathcal{U}_n$-twisted isometry is nuclear. We exhibit concrete analytic models of $\mathcal{U}_n$-twisted isometries, and establish connections between unitary equivalence classes of the irreducible representations of the $C^*$-algebras generated by $\mathcal{U}_n$-twisted isometries and the unitary equivalence classes of the non-zero irreducible representations of twisted noncommutative tori. Our motivation of $\mathcal{U}_n$-twisted isometries stems from the classical rotation $C^*$-algebras, Heisenberg group $C^*$-algebras, and a recent work of de Jeu and Pinto.