On split exact sequences and KK-equivalences of amplified graph C*-algebras

TL;DR

This paper develops a methodology for constructing split exact sequences in amplified graph C*-algebras, enabling explicit KK-equivalences with finite-dimensional algebras, including quantum and classical projective spaces.

Contribution

It introduces a general approach for building split exact sequences in amplified graph C*-algebras, facilitating explicit KK-equivalences with finite-dimensional and quantum spaces.

Findings

Constructed split exact sequences for amplified graph C*-algebras with sinks.

Established explicit KK-equivalences with ^N for various C*-algebras.

Provided an explicit KK-equivalence between classical and quantum projective spaces.

Abstract

We give a general methodology for constructing split exact sequences of amplified graph C*-algebras with sinks. This in turn allows us to construct explicit KK-equivalences with $\mathbb{C}^N$ for a large class of C*-algebras, including the quantum Grassmannian $\mathrm{Gr}_q(2,4)$. We discuss compatibility with known (quantum) CW-constructions and give an explicit KK-equivalence between the classical and quantum projective spaces $\mathbb{C}P^1$ and $\mathbb{C}P_q^1$.