# On Bergman-Toeplitz operators with commutative symbol algebras

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## Abstract

Let\(\mathbb{D}\) be the unit disk inℂ,\(\mathcal{A}^2 (\mathbb{D})\) be the Bergman space, consisting of all analytic functions from\(L_2 (\mathbb{D})\), and\(B_\mathbb{D} \) be the Bergman projection of\(L_2 (\mathbb{D})\) onto\(\mathcal{A}^2 (\mathbb{D})\). We construct*C*^{*}-algebras\(\mathcal{A} \subset L_\infty (\mathbb{D})\), for functions of which the commutator of Toeplitz operators [*T*_{ a },*T*_{ b }]=*T*_{ a }*T*_{ b }*−T*_{ b }*T*_{ a } is compact, and, at the same time, the semi-commutator [*T*_{ a },*T*_{ b })=*T*_{ a }*T*_{ b }*−T*_{ ab } is not compact.

It is proved, that for each finite set ∧=*〈n*_{0},*n*_{1}, ...,*n*_{ m }*〉*, where 1=*n*_{0}*<n*_{1}*<...<n*_{ m }≤∞, and*n*_{ k } ∈ℕ∪ {∞}, there are algebras\(\mathcal{A}_\Lambda \) of the above type, such that the symbol algebras Sym\(\mathcal{T}(\mathcal{A}_\Lambda )\) of Toeplitz operator algebras\(\mathcal{T}(\mathcal{A}_\Lambda )\) are*commutative*, while the symbol algebras Sym\(\mathcal{R}(\mathcal{A}_\Lambda ,B_\mathbb{D} )\) of the algebras\(\mathcal{R}(\mathcal{A}_\Lambda ,B_\mathbb{D} )\), generated by multiplication operators\(a \in \mathcal{A}_\Lambda \) and\(B_\mathbb{D} \), have*irreducible representations exactly of dimensions n*_{0},*n*_{1},*..., n*_{ m }.

## AMS Classification

47B35 47D25## Preview

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