zbMATH — the first resource for mathematics

Toeplitz \(C^*\)-algebras and several complex variables. (English) Zbl 0833.47019
Curto, Raúl E. (ed.) et al., Multivariable operator theory. A joint summer research conference on multivariable operator theory, July 10-18, 1993, University of Washington, Seattle, WA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 185, 339-346 (1995).
The author presents a brief but deep survey of the \(C^*\)-algebraic properties of Toeplitz operators on a domain \(\Omega \subseteq \mathbb{C}^n\). The main object of study is the \(C^*\)-algebra \({\mathcal I} (\partial \Omega)\) generated by Toeplitz operators on a canonical Hilbert space of holomorphic functions on \(\Omega\), with symbols in a natural function algebra over \(\Omega\).
The main problems which are discussed here are the following:
When does the commutator ideal \({\mathcal C}\) in \({\mathcal I} (\partial \Omega)\) coincide with the ideal of all compact operators \({\mathcal K}\)?
What is the structure of \({\mathcal C}/K\)?
What is its relationship with \(\partial \Omega\)?
Theorems which give answers to these questions for strongly pseudoconvex domains are formulated and the structure of the commutator ideal \({\mathcal C}\) is described for pseudoconvex Reinhart domains in \(\mathbb{C}^n\) and more general \(K\)-circular domains where \(K\) is a real compact Lie group.
For the entire collection see [Zbl 0819.00022].
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
46L05 General theory of \(C^*\)-algebras
32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010)
32T99 Pseudoconvex domains