Examples of *-commuting maps

We introduce the concept of a 1-coaligned $k$-graph and prove that the shift maps of a $k$-graph pairwise *-commute if and only if the $k$-graph is 1-coaligned. We then prove that for 2-graphs $Λ$ generated from basic data *-commuting shift maps is equivalent to a condition that implies that $C^*(Λ)$ is simple and purely infinite. We then consider full shift spaces and introduce a condition on a block map which ensures the associated sliding block code *-commutes with the shift.

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