Permutative representations of the Cuntz-Krieger algebras

We generalize permutative representations of the Cuntz algebras for the \cka
$\text{coa}$ for any $A$. We characterize cyclic permutative representations by notions of cycle and chain, and show their existence and uniqueness. We show necessary and sufficient conditions for their irreducibility and equivalence. In consequence, we have a complete classification of permutative representations of $\text{coa}$. Furthermore we show decomposition formulae and that the uniqueness of irreducible decomposition holds for permutative representation.