Differential norms and Rieffel algebras

We develop criteria to guarantee uniqueness of the $${\rm C}^*$$-norm on a $$*$$-algebra $$\mathcal {B}$$. Nontrivial examples are provided by the noncommutative algebras of $$\mathcal {C}$$-valued functions $$\mathcal {S}_J^\mathcal {C}(\mathbb {R}^n)$$ and $$\mathcal {B}_J^\mathcal {C}(\mathbb {R}^n)$$ defined by M.A. Rieffel via a deformation quantization procedure, where $$\mathcal {C}$$ is a $${\rm C}^*$$-algebra and $$J$$ is a skew-symmetric linear transformation on $$\mathbb {R}^n$$ with respect to which the usual pointwise product is deformed. In the process, we prove that the Fréchet $$*$$-algebra topology of $$\mathcal {B}_J^\mathcal {C}(\mathbb {R}^n)$$ can be generated by a sequence of submultiplicative $$*$$-norms and that, if $$\mathcal {C}$$ is unital, this algebra is closed under the $${\rm C}^\infty$$-functional calculus of its $${\rm C}^*$$-completion. We also show that the algebras $$\mathcal {S}_J^\mathcal {C}(\mathbb {R}^n)$$ and $$\mathcal {B}_J^\mathcal {C}(\mathbb {R}^n)$$ are spectrally invariant in their respective $${\rm C}^*$$-completions, when $$\mathcal {C}$$ is unital. As a corollary of our results, we obtain simple proofs of certain estimates in $$\mathcal {B}_J^\mathcal {C}(\mathbb {R}^n)$$.