We study the noncommutative topology of the $C^*$-algebras $C(\mathbb{ C}P_q^{ n})$ of the quantum projective spaces within the framework of Kasparov's bivariant K-theory. In particular, we construct an explicit KK-equivalence with the commutative algebra $\mathbb{ C}^{ n+1}$. Our construction relies on showing that the extension of $C^*$-algebras relating two quantum projective spaces of successive dimensions admits a splitting, which we can describe explicitly using graph algebra techniques.
Categories: math.OA math.KT math.QA