Differential norms and Rieffel algebras

Abstract

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