On the classification of $C^{\ast }$-algebras of twisted isometries with finite dimensional wandering spaces

Let $m,n\in {\mathbb{N}}_0$, and let $X$ be a closed subset of ${\mathbb{T}}^{\left(\genfrac{}{}{0px}{}{m+n}{2}\right)}$. We define $C_X^{m,n}$ to be the universal $C^{\ast }$-algebra among those generated by $m$ unitaries and $n$ isometries satisfying doubly twisted commutation relations with respect to a twist $\mathcal{U}=\{U_{ij}{\}}_{1\le i<j\le m+n}$ of commuting unitaries having joint spectrum $X$. We provide a complete list of the irreducible representations of $C_X^{m,n}$ up to unitary equivalence and, under a denseness assumption on $X$, explicitly construct a faithful representation of $C_X^{m,n}$. Under the same assumption, we also give a necessary and sufficient condition on a fixed tuple $\mathcal{U}$ of commuting unitaries with joint spectrum $X$ for the existence of a universal tuple of $\mathcal{U}$-doubly twisted isometries. For $X={\mathbb{T}}^{\left(\genfrac{}{}{0px}{}{m+n}{2}\right)}$, we compute the $K$-groups of $C_X^{m,n}$. We further classify the $C^{\ast }$-algebras generated by a pair of doubly twisted isometries with a fixed parameter $\theta \in \mathbb{R}\setminus \mathbb{Q}$, whose wandering spaces are finite-dimensional. Finally, for a fixed unitary $U$, we classify all the $C^{\ast }$-algebras generated by a pair of $U$-doubly twisted isometries with finite-dimensional wandering spaces.