Equivariant $K$-theory of cellular toric varieties

We describe the $T_{\rm comp}$-equivariant topological $K$-ring of a complete $T$-cellular toric variety. We further show that $K_{T_{\textrm{comp}}}^0(X)$ is isomorphic as an $R(T_{\rm comp})$-algebra to the ring of piecewise Laurent polynomial functions on the associated fan denoted ${\rm PLP}(\Delta )$. Furthermore, we compute a basis for $K_{T_{\rm comp}}^0(X)$ as an $R(T_{\rm comp})$-module, and multiplicative structure constants with respect to this basis.