The K-theory of the triple-Toeplitz deformation of the complex projective plane

TL;DR

This paper computes the K-theory of a specific noncommutative deformation of the complex projective plane using a family of $C^*$-algebra epimorphisms satisfying a cocycle condition, exemplified by the triple-Toeplitz deformation.

Contribution

It introduces a method to compute K-groups of multi-pullback $C^*$-algebras under cocycle conditions, applied to the triple-Toeplitz deformation of $ ext{CP}^2$.

Findings

K-groups of the multi-pullback algebra are explicitly computed.

The triple-Toeplitz deformation of $ ext{CP}^2$ has determined K-theory.

Method generalizes to other noncommutative deformations.

Abstract

We consider a family $\pi^i_j\colon B_i\rightarrow B_{ij}=B_{ji}$, $i,j\in \{1,2,3\}$, $i\neq j$, of $C^*$-epimorphisms assuming that it satisfies the cocycle condition. Then we show how to compute the $K$-groups of the multi-pullback $C^*$-algebra of such a family, and examplify it in the case of the triple-Toeplitz deformation of $\mathbb{C}P^2$.