Noncommutative field theory on ${\mathbb{R}}_{\lambda }^3$

We consider the noncommutative space ${\mathbb{R}}_{\lambda }^3$, a deformation of the algebra of functions on ${\mathbb{R}}^3$ which yields a foliation of ${\mathbb{R}}^3$ into fuzzy spheres. We first review the construction of a natural matrix basis adapted to ${\mathbb{R}}_{\lambda }^3$. We thus consider the problem of defining a new Laplacian operator for the deformed algebra. We propose an operator which is not of Jacobi type. The implication for field theory of the new Laplacian is briefly discussed.