Rank-two Milnor idempotents for the multipullback quantum complex projective plane

Abstract

The $K_0$-group of the C*-algebra of multipullback quantum complex projective plane is known to be ${\mathbb{Z}}^3$, with one generator given by the C*-algebra itself, one given by the section module of the noncommutative (dual) tautological line bundle, and one given by the Milnor module associated to a generator of the $K_1$-group of the C*-algebra of Calow-Matthes quantum 3-sphere. Herein we prove that these Milnor modules are isomorphic either to the section module of a noncommutative vector bundle associated to the $S{U}_q(2)$-prolongation of the Heegaard quantum 5-sphere $S_H^5$ viewed as a $U(1)$-quantum principal bundle, or to a complement of this module in the rank-four free module. Finally, we demonstrate that one of the above Milnor modules always splits into the direct sum of the rank-one free module and a rank-one non-free projective module that is \emph{not} associated with $S_H^5$.

This paper has been superseded by arXiv:2512.08304, which is not just an update, or a new version. It contains new research obtain by different authors