  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
$\mathcal{U}_n = \{U_{ij}\}_{i\neq j} \subseteq \mathcal{B}(\mathcal{H})$,
where $U_{ji} := U_{ij}^*$, $1 \leq i < j \leq n$. An $n$-tuple of isometries
$V = (V_1, \ldots ,V_n)$ on $\mathcal{H}$ is called $\mathcal{U}_n$-twisted
isometry 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.
  We prove that the universal $C^*$-algebra generated by
$\mathcal{U}_n$-twisted isometry is nuclear. The universal $C^*$-algebra
generated by an $n$-tuple of $\mathcal{U}_n$-twisted unitaries is called the
generalized noncommutative $n$-torus. 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 generalized
noncommutative tori. Our motivation stems from the classical rotation
$C^*$-algebras, Heisenberg group $C^*$-algebras, and a recent work of de Jeu
and Pinto.
