Complete positivity of Rieffel’s deformation quantization by actions of ${\mathbb{R}}^d$

Abstract

In this paper we consider ${\mathrm{C}}^{\ast }$-algebraic deformations by actions of ${\mathbb{R}}^d$ à la Rieffel and show that every state of the undeformed algebra can be deformed into a state of the deformed algebra in the sense of a continuous field of states. The construction is explicit and involves a convolution operator with a particular Gauß function.