On algebras generated by Toeplitz operators and their representations

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• 作者：Wolfram Bauer^a ; ^{bauer@math.uni-hannover.de" class="auth_mail" title="E-mail the corresponding author} ; Nikolai Vasilevski^b ; ^{nvasilev@math.cinvestav.mx" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 47B35 ; 47L80 ; secondary ; 32A36
• 刊名：Journal of Functional Analysis
• 出版年：2017
• 出版时间：15 January 2017
• 年：2017
• 卷：272
• 期：2
• 页码：705-737
• 全文大小：568 K

文摘

We study Banach and C^⁎$C^{⁎}$-algebras generated by Toeplitz operators acting on weighted Bergman spaces ${\mathcal{A}}_{\lambda }^2({\mathbb{B}}^2)$ over the complex unit ball fbe4bf2f391edea20ae2674a6" title="Click to view the MathML source">B²⊂C²${\mathbb{B}}^2\subset {\mathbb{C}}^2$. Our key point is an orthogonal decomposition of ${\mathcal{A}}_{\lambda }^2({\mathbb{B}}^2)$ into a countable sum of infinite dimensional spaces, each one of which can be identified with a differently weighted Bergman space ${\mathcal{A}}_{\mu }^2(\mathbb{D})$ over the complex unit disk D$\mathbb{D}$. Moreover, all elements of the above algebras leave each of the summands in the above decomposition invariant and their restriction to each level acts as a compact perturbation of a Toeplitz operator on ${\mathcal{A}}_{\mu }^2(\mathbb{D})$.

The symbols of the generating Toeplitz operators are chosen to be suitable extensions to B²${\mathbb{B}}^2$ of families S$\mathcal{S}$ of bounded functions on D$\mathbb{D}$. Symbol classes S$\mathcal{S}$ that generate important classical commutative and non-commutative Toeplitz algebras in $\mathcal{L}({\mathcal{A}}_{\mu }^2(\mathbb{D}))$ are of particular interest. In this paper we discuss various examples. In the case of $\mathcal{S}=C(\bar{\mathbb{D}})$ and $\mathcal{S}=C(\bar{\mathbb{D}})\otimes L_{\infty }(0,1)$ we characterize all irreducible representations of the resulting Toeplitz operator C^⁎$C^{⁎}$-algebras. Their Calkin algebras are described and index formulas are provided.