Toeplitz algebra and Symbol map via Berezin transform on $H^2(\mathbb{D}^n)$
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914801593090048 |
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| author | Javed, Mo Maji, Amit |
| author_facet | Javed, Mo Maji, Amit |
| contents | Let $\mathscr{T}(L^{\infty}(\mathbb{T}))$ be the Toeplitz algebra, that is, the $C^*$-algebra generated by the set $\{T_ϕ : ϕ\in L^{\infty}(\mathbb{T})\}$. Douglas's theorem on symbol map states that there exists a $C^*$-algebra homomorphism from $\mathscr{T}(L^{\infty}(\mathbb{T}))$ onto $L^{\infty}(\mathbb{T})$ such that $T_ϕ\mapsto ϕ$ and the kernel of the homomorphism coincides with commutator ideal in $\mathscr{T}(L^{\infty}(\mathbb{T}))$. In this paper, we use the Berezin transform to study results akin to Douglas's theorem for operators on the Hardy space $H^2(\mathbb{D}^n)$ over the open unit polydisc $\mathbb{D}^n$ for $n\geq 1$. We further obtain a class of bigger $C^*$-algebras than the Toeplitz algebra $\mathscr{T}(L^{\infty}(\mathbb{T}^n))$ for which the analog of symbol map still holds true. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_10967 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Toeplitz algebra and Symbol map via Berezin transform on $H^2(\mathbb{D}^n)$ Javed, Mo Maji, Amit Functional Analysis Complex Variables Operator Algebras 47B35, 47A13, 32A65 Let $\mathscr{T}(L^{\infty}(\mathbb{T}))$ be the Toeplitz algebra, that is, the $C^*$-algebra generated by the set $\{T_ϕ : ϕ\in L^{\infty}(\mathbb{T})\}$. Douglas's theorem on symbol map states that there exists a $C^*$-algebra homomorphism from $\mathscr{T}(L^{\infty}(\mathbb{T}))$ onto $L^{\infty}(\mathbb{T})$ such that $T_ϕ\mapsto ϕ$ and the kernel of the homomorphism coincides with commutator ideal in $\mathscr{T}(L^{\infty}(\mathbb{T}))$. In this paper, we use the Berezin transform to study results akin to Douglas's theorem for operators on the Hardy space $H^2(\mathbb{D}^n)$ over the open unit polydisc $\mathbb{D}^n$ for $n\geq 1$. We further obtain a class of bigger $C^*$-algebras than the Toeplitz algebra $\mathscr{T}(L^{\infty}(\mathbb{T}^n))$ for which the analog of symbol map still holds true. |
| title | Toeplitz algebra and Symbol map via Berezin transform on $H^2(\mathbb{D}^n)$ |
| topic | Functional Analysis Complex Variables Operator Algebras 47B35, 47A13, 32A65 |
| url | https://arxiv.org/abs/2405.10967 |