Differential norms and Rieffel algebras

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