A unitary invariant for $C^*$-algebras generated by isometries with a twisted commutation relation

Abstract

We investigate the classification of $C^*$-algebras generated by pairs of isometries $(V_1, V_2)$ satisfying $V_1V_2=qV_2V_1$, where $q=e^{2\pi i\theta}$ with $\theta$ irrational, up to unitary equivalence. Using a functional model that realizes pure $q$-commuting pairs on vector-valued Hardy spaces in terms of a uniquely associated triple $(\mathcal{F}, P, U)$ consisting of Hilbert space $\mathcal{F}$, an orthogonal projection $P$ and a unitary operator $U$, we analyze the internal structure of $C^*(V_1,V_2)$ via an extension of the $C^*$-algebra generated by its minimal $q$-commuting unitary extension by the defect ideal $\mathcal{I}_{\mathcal{D}} =\langle [V_1^*, V_1],[V_2^*, V_2]\rangle$. For pairs that are essentially doubly $q$-commuting, we show that $PUP|_{\operatorname{ran} P}$ is semi-Fredholm and prove that $|\operatorname{ind} PUP|_{\operatorname{ran} P}|$ is invariant under unitary equivalence of the corresponding $C^*$-algebras, thereby providing an explicit criterion for distinguishing non-equivalent pairs. This work extends the seminal 1978 results of Berger, Coburn, and Lebow for commuting isometries, while addressing the key technical nuances of this noncommutative setting: most notably, navigating the failure of Gelfand theory due to noncommutativity and replacing the classical Toeplitz extension by an extension of the noncommutative torus by the ideal of compact operators.