Differential Norms and Rieffel Algebras

Abstract

We develop criteria to guarantee uniqueness of the C${}^{\ast }$-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 C${}^{\ast }$-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 C${}^{\infty }$-functional calculus of its C${}^{\ast }$-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 C${}^{\ast }$-completions, when $\mathcal{C}$ is unital. As a corollary of our results, we obtain simple proofs of certain estimates in ${\mathcal{B}}_J^{\mathcal{C}}({\mathbb{R}}^n)$.

Minor changes only. Final version to appear in Mathematische Nachrichten