Split extensions and KK-equivalences for quantum projective spaces

TL;DR

This paper investigates the noncommutative topology of quantum projective spaces' $C^*$-algebras, establishing explicit KK-equivalences with commutative algebras through extension splitting and graph algebra techniques.

Contribution

It constructs explicit KK-equivalences between quantum and classical projective spaces, revealing their topological relationship via extension splitting.

Findings

Established KK-equivalence with classical algebras

Explicit construction of extension splitting

Applied graph algebra techniques

Abstract

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.