A unitary invariant for $C^*$-algebras generated by isometries with a twisted commutation relation
Year: 2026
Type: preprint
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 $q=e^{2πiθ}$ with $θ$ 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.
Source: arXiv (Cornell University)
Authors: Ghosh, Sourav, Sau, Haripada
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214658445