Differential Norms and Rieffel Algebras

TL;DR

This paper establishes criteria for the uniqueness of C*-norms on *-algebras, focusing on noncommutative Rieffel deformed algebras, and proves their spectral invariance and functional calculus properties.

Contribution

It introduces new criteria for C*-norm uniqueness on *-algebras and demonstrates spectral invariance and functional calculus closure for Rieffel deformed algebras.

Findings

Spectral invariance of Rieffel deformed algebras when  is unital

Generation of algebra topology by submultiplicative *-norms

Closure under C^-functional calculus in C*-completion

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 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$^\infty$-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 $\mathcal{B}_J^\mathcal{C}(\mathbb{R}^n)$.