On the classification of $C^*$-algebras of twisted isometries with finite dimensional wandering spaces

Abstract: Let ( m, n \in \mathbb{N}0 ), and let ( X ) be a closed subset of ( \mathbb{T}^{\binom{m+n}{2}} ). We define ( C^{m,n}_X ) to be the universal ( C^* )-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 \leq i < j \leq m+n} ) of commuting unitaries having joint spectrum ( X ). We provide a complete list of the irreducible representations of ( C^{m,n}_X ) up to unitary equivalence and, under a denseness assumption on ( X ), explicitly construct a faithful representation of ( C^{m,n}_X ). 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}^{\binom{m+n}{2}} ), we compute the ( K )-groups of ( C^{m,n}_X ). We further classify the ( C^* )-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^* )-algebras generated by a pair of ( U )-doubly twisted isometries with finite-dimensional wandering spaces.