Differential Norms and Rieffel Algebras
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| author | Cabral, Rodrigo A. H. M. Forger, Michael Melo, Severino T. |
| author_facet | Cabral, Rodrigo A. H. M. Forger, Michael Melo, Severino T. |
| contents | 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 $\mathcal{B}_J^\mathcal{C}(\mathbb{R}^n)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_02380 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Differential Norms and Rieffel Algebras Cabral, Rodrigo A. H. M. Forger, Michael Melo, Severino T. Operator Algebras Functional Analysis Primary: 46H35, 47G30, 46L87. Secondary: 46L52, 46L08, 43A65 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 $\mathcal{B}_J^\mathcal{C}(\mathbb{R}^n)$. |
| title | Differential Norms and Rieffel Algebras |
| topic | Operator Algebras Functional Analysis Primary: 46H35, 47G30, 46L87. Secondary: 46L52, 46L08, 43A65 |
| url | https://arxiv.org/abs/2110.02380 |