Differential norms and Rieffel algebras

Rodrigo A. H. M. Cabral, Michael Forger and Severino T. Melo

Mathematische Nachrichten, 2025, vol. 298, issue 7, 2177-2203

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

Date: 2025
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