Equivariant $K$-theory of cellular toric varieties

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