## Algebras of Toeplitz operators on the n-dimensional unit ball
Bauer, W., Hagger, R. and Vasilevski, N.
(2019)
It is advisable to refer to the publisher's version if you intend to cite from this work. See Guidance on citing. To link to this item DOI: 10.1007/s11785-018-0837-y ## Abstract/SummaryWe study $C^*$-algebras generated by Toeplitz operators acting on the standard weighted Bergman space $\mathcal{A}_{\lambda}^2(\mathbb{B}^n)$ over the unit ball $\mathbb{B}^n$ in $\mathbb{C}^n$. The symbols $f_{ac}$ of generating operators are assumed to be of a certain product type, see (\ref{Introduction_form_of_the_symbol}). By choosing $a$ and $c$ in different function algebras $\mathcal{S}_a$ and $\mathcal{S}_c$ over lower dimensional unit balls $\mathbb{B}^{\ell}$ and $\mathbb{B}^{n-\ell}$, respectively, and by assuming the invariance of $a\in \mathcal{S}_a$ under some torus action we obtain $C^*$-algebras $\boldsymbol{\mathcal{T}}_{\lambda}(\mathcal{S}_a, \mathcal{S}_c)$ whose structural properties can be described. In the case of $k$-quasi-radial functions $\mathcal{S}_a$ and bounded uniformly continuous or vanishing oscillation symbols $\mathcal{S}_c$ we describe the structure of elements from the algebra $\boldsymbol{\mathcal{T}}_{\lambda}(\mathcal{S}_a, \mathcal{S}_c)$, derive a list of irreducible representations of $\boldsymbol{\mathcal{T}}_{\lambda}(\mathcal{S}_a, \mathcal{S}_c)$, and prove completeness of this list in some cases. Some of these representations originate from a ``quantization effect'', induced by the representation of $\mathcal{A}_{\lambda}^2(\mathbb{B}^n)$ as the direct sum of Bergman spaces over a lower dimensional unit ball with growing weight parameter. As an application we derive the essential spectrum and index formulas for matrix-valued operators.
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