Nonstandard Quantum Complex Projective Line

Abstract

In our attempt to explore how the quantum nonstandard complex projective spaces ℂ𝘗ⁿq,c studied by Korogodsky, Vaksman, Dijkhuizen, and Noumi are related to those arising from the geometrically constructed Bohr-Sommerfeld groupoids by Bonechi, Ciccoli, Qiu, Staffolani, and Tarlini, we were led to establish the known identification of 𝐶(ℂ𝘗¹q,c) with the pull-back of two copies of the Toeplitz 𝐶*-algebra along the symbol map in a more direct way via an operator theoretic analysis, which also provides some interesting non-obvious details, such as a prominent generator of 𝐶(ℂ𝘗¹q,c) being a concrete weighted double shift.