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

@misc{ghosh2026unitary,
  title = {{A unitary invariant for $C^*$-algebras generated by isometries with a twisted commutation relation}},
  author = {Sourav Ghosh and Haripada Sau},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.31220},
  eprint = {2609.31220},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.31220},
  note = {CC0},
}

TY  - UNPB
TI  - A unitary invariant for C^*-algebras generated by isometries with a twisted commutation relation
AU  - Sourav Ghosh
AU  - Haripada Sau
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.31220
UR  - https://arxiv.org/abs/2609.31220
AB  - 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 (ℱ, P, U) consisting of Hilbert space 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 ℐ_𝒟 =⟨ [V_1^*, V_1],[V_2^*, V_2]⟩. For pairs that are essentially doubly q-commuting, we show that PUP|_ran P is semi-Fredholm and prove that |ind PUP|_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.
ER  -