Quantization and the Resolvent Algebra
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arXiv
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| Format: | Preprint |
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2019
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| _version_ | 1866910733630963712 |
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| author | van Nuland, Teun D. H. |
| author_facet | van Nuland, Teun D. H. |
| contents | We introduce a novel commutative C*-algebra $C_\mathcal{R}(X)$ of functions on a symplectic vector space $(X,σ)$ admitting a complex structure, along with a strict deformation quantization that maps a dense subalgebra of $C_\mathcal{R}(X)$ to the resolvent algebra introduced by Buchholz and Grundling [JFA, 2008]. The associated quantization map is a field-theoretical Weyl quantization compatible with the work of Binz, Honegger and Rieckers [AHPO, 2004]. We also define a Berezin-type quantization map on all of $C_\mathcal{R}(X)$, which continuously and injectively maps it onto a dense subset of the resolvent algebra.
The commutative resolvent algebra $C_\mathcal{R}(X)$, generally defined on a real inner product space $X$, intimately depends on the finite dimensional subspaces of $X$. We thoroughly analyze the structure of this algebra in the finite dimensional case by giving a characterization of its elements and by computing its Gelfand spectrum. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1903_04819 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Quantization and the Resolvent Algebra van Nuland, Teun D. H. Functional Analysis We introduce a novel commutative C*-algebra $C_\mathcal{R}(X)$ of functions on a symplectic vector space $(X,σ)$ admitting a complex structure, along with a strict deformation quantization that maps a dense subalgebra of $C_\mathcal{R}(X)$ to the resolvent algebra introduced by Buchholz and Grundling [JFA, 2008]. The associated quantization map is a field-theoretical Weyl quantization compatible with the work of Binz, Honegger and Rieckers [AHPO, 2004]. We also define a Berezin-type quantization map on all of $C_\mathcal{R}(X)$, which continuously and injectively maps it onto a dense subset of the resolvent algebra. The commutative resolvent algebra $C_\mathcal{R}(X)$, generally defined on a real inner product space $X$, intimately depends on the finite dimensional subspaces of $X$. We thoroughly analyze the structure of this algebra in the finite dimensional case by giving a characterization of its elements and by computing its Gelfand spectrum. |
| title | Quantization and the Resolvent Algebra |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/1903.04819 |