Equivariant Index Theorem on ${\mathbb{R}}^n$ in the Context of Continuous Fields of $C^{\ast }$-algebras

We prove an equivariant index theorem on the Euclidean space using a continuous field of $C^{\ast }$-algebras. This generalizes the work of Elliott, Natsume and Nest, which is a special case of the algebraic index theorem by Nest-Tsygan. Using our formula, the equivariant index of the Bott-Dirac operator on ${\mathbb{R}}^{2n}$ can be explicitly calculated.