On the magic square C*-algebra of size 4

TL;DR

This paper analyzes the structure of the size 4 magic square C*-algebra, revealing its isomorphism to a fixed point algebra of a matrix algebra over real projective 3-space, and computes its K-groups.

Contribution

It establishes a concrete realization of the magic square C*-algebra of size 4 as a fixed point algebra, enabling explicit K-theory computations.

Findings

A twisted crossed product of $A(4)$ is isomorphic to $M_4(C(\,\mathbb{R} P^3))$

$A(4)$ is isomorphic to the fixed point algebra of $M_4(C(\,\mathbb{R} P^3))$ under a specific action

Computed the K-groups of $A(4)$ and identified their generators

Abstract

In this paper, we investigate the structure of the magic square C*-algebra $A(4)$ of size 4. We show that a certain twisted crossed product of $A(4)$ is isomorphic to the homogeneous C*-algebra $M_4(C(\mathbb{R} P^3))$. Using this result, we show that $A(4)$ is isomorphic to the fixed point algebra of $M_4(C(\mathbb{R} P^3))$ by a certain action. From this concrete realization of $A(4)$, we compute the K-groups of $A(4)$ and their generators.