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We develop criteria to guarantee uniqueness of the C$^*$-norm on a *-algebra$\mathcal{B}$. Nontrivial examples are provided by the noncommutative algebrasof $\mathcal{C}$-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 adeformation quantization procedure, where $\mathcal{C}$ is a C$^*$-algebra and$J$ is a skew-symmetric linear transformation on $\mathbb{R}^n$ with respect towhich the usual pointwise product is deformed. In the process, we prove thatthe Fr\'echet *-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$^\infty$-functionalcalculus of its 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 theirrespective C$^*$-completions, when $\mathcal{C}$ is unital. As a corollary ofour results, we obtain simple proofs of certain estimates in$\mathcal{B}_J^\mathcal{C}(\mathbb{R}^n)$.

Differential Norms and Rieffel Algebras