Differential Norms and Rieffel Algebras
Rodrigo A. H. M. Cabral, Michael Forger, Severino T. Melo
TL;DR
The paper develops criteria ensuring a $*$-algebra $\mathcal{B}$ has a unique C$^*$-norm when realized as a dense subalgebra of a C$^*$-algebra $\mathcal{A}$, with a key result that closure under the $C^{\infty}$-functional calculus of $\mathcal{A}$ implies uniqueness. It then applies these ideas to Rieffel’s noncommutative function algebras $\mathcal{S}_J^\mathcal{C}(\mathbb{R}^n)$ and $\mathcal{B}_J^\mathcal{C}(\mathbb{R}^n)$ obtained via a deformation product $\times_J$, realized on a Hilbert $\mathcal{C}$-module $E_n$ through the operators $L_f$. The section constructs a differential norm generating the natural Fréchet topology, proves a Calderón–Vaillancourt-type inequality for Hilbert C$^*$-modules, and establishes spectral invariance and K-theory stability for these deformed algebras, with $J=0$ recovering the undeformed sup-norm scenario. The results provide a unified framework for C$^*$-norm uniqueness across smooth/deformed algebras and have implications for pseudodifferential operators and noncommutative geometry.
Abstract
We develop criteria to guarantee uniqueness of the 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 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 C$^\infty$-functional calculus 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 their respective C$^*$-completions, when $\mathcal{C}$ is unital. As a corollary of our results, we obtain simple proofs of certain estimates in $\mathcal{B}_J^\mathcal{C}(\mathbb{R}^n)$.