Bases of the quantum cluster algebra of the Kronecker quiver

We construct bar-invariant $\mathbb{Z}[q^{\pm 1/2}]-$bases of the quantum cluster algebra of the Kronecker quiver which are quantum analogues of the canonical basis, semicanonical basis and dual semicanonical basis of the cluster algebra of the Kronecker quiver in the sense of \cite{sherzel},\cite{calzel} and \cite{gls} respectively. As a byproduct, we prove the positivity of the elements in these bases.