Split extensions and KK-equivalences for quantum projective spaces

Abstract

We study the noncommutative topology of the $C^{\ast }$-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 $C^{\ast }$-algebras relating two quantum projective spaces of successive dimensions admits a splitting, which we can describe explicitly using graph algebra techniques.

Section 6 has been divided into two subsections, where we in 6.1 go through the construction of the KK-equivalence in larger generality. Minor changes and corrections