[Paper Review] Ideals in Toeplitz Algebras

Ronald G. Douglas|ArXiv.org|Sep 26, 2003

Advanced Topics in Algebra6 references3 citations

TL;DR

This paper determines the ideal structure of the Toeplitz $C^*$-algebra on the bidisk by analyzing the kernel of symbol maps and showing that the intersection of the kernels of three symbol maps on $\mathfrak{T} \otimes \mathfrak{T} \otimes \mathfrak{T}$ is $\mathfrak{K} \otimes \mathfrak{K} \otimes \mathfrak{K}$, which characterizes the minimal nontrivial ideal. The result extends to higher-rank domains via analogous symbol map decompositions.

ABSTRACT

We determine the ideal structure of the Toeplitz C*-algebra on the bidisk.

Motivation & Objective

• To determine the ideal structure of the Toeplitz $C^*$-algebra on the bidisk, particularly focusing on the interplay between module multiplication operators and quotient Hilbert modules.
• To investigate whether a $C^*$-algebra homomorphism exists between the algebra generated by multiplication operators on a Hardy module and that on its quotient module.
• To analyze the kernel of symbol maps on tensor products of Toeplitz algebras to characterize the ideals in the $C^*$-algebra of the bidisk.
• To extend the characterization of ideals in the Toeplitz algebra of the bidisk to higher-dimensional domains using multilinear symbol maps.
• To demonstrate that the intersection of the kernels of three symbol maps on $\mathfrak{T} \otimes \mathfrak{T} \otimes \mathfrak{T}$ is $\mathfrak{K} \otimes \mathfrak{K} \otimes \mathfrak{K}$, which is the minimal nontrivial ideal.

Proposed method

• Uses the symbol map $\Sigma: \mathfrak{T} \otimes \mathfrak{T} \to C((\partial\mathbb{D})^2)$ to analyze the structure of the Toeplitz $C^*$-algebra on the bidisk.
• Applies exact sequences involving $C(\partial\mathbb{D}, \mathfrak{K})$ and $C(\partial\mathbb{D}, \mathfrak{T})$ to characterize elements in the kernel of the symbol map.
• Employs the tensor product structure $\mathfrak{T} \otimes \mathfrak{T}$ and the maps $\alpha_z$ and $\beta_w$ to decompose elements in the kernel and verify surjectivity.
• Uses the exactness of the sequence $0 \to C(\partial\mathbb{D}, \mathfrak{K}) \to C(\partial\mathbb{D}, \mathfrak{T}) \to C((\partial\mathbb{D})^2) \to 0$ to show that elements in the kernel are compact in each fiber.
• Applies the same technique to $\mathfrak{T} \otimes \mathfrak{T} \otimes \mathfrak{T}$, using three symbol maps $\Sigma_1, \Sigma_2, \Sigma_3$ to characterize the intersection of their kernels.
• Demonstrates that the intersection of $\ker(\sigma \otimes 1)$ and $\ker(1 \otimes \sigma)$ in $\mathfrak{T} \otimes \mathfrak{T}$ is $\mathfrak{K} \otimes \mathfrak{K}$, which is contained in every nontrivial ideal.

Experimental results

Research questions

• RQ1 What is the ideal structure of the Toeplitz $C^*$-algebra on the bidisk, particularly in relation to quotient modules of the Hardy space?
• RQ2 Can a $C^*$-algebra homomorphism be constructed from the Toeplitz algebra of a Hardy module to that of its quotient module?
• RQ3 How do the kernels of symbol maps on $\mathfrak{T} \otimes \mathfrak{T}$ interact to characterize the minimal nontrivial ideal?
• RQ4 What is the structure of the intersection of the kernels of three symbol maps on $\mathfrak{T} \otimes \mathfrak{T} \otimes \mathfrak{T}$?
• RQ5 Can the ideal structure of the Toeplitz $C^*$-algebra on higher-dimensional domains be described analogously using multilinear symbol maps?

Key findings

• The ideal structure of the Toeplitz $C^*$-algebra on the bidisk is fully characterized by the intersection of the kernels of three symbol maps, which is $\mathfrak{K} \otimes \mathfrak{K}$.
• The intersection of $\ker(\sigma \otimes 1)$ and $\ker(1 \otimes \sigma)$ in $\mathfrak{T} \otimes \mathfrak{T}$ is $\mathfrak{K} \otimes \mathfrak{K}$, which is the minimal nontrivial ideal in the algebra.
• For the triple tensor product $\mathfrak{T} \otimes \mathfrak{T} \otimes \mathfrak{T}$, the intersection of the kernels of the three symbol maps $\Sigma_1, \Sigma_2, \Sigma_3$ is $\mathfrak{K} \otimes \mathfrak{K} \otimes \mathfrak{K}$.
• The minimal nontrivial ideal in $\mathfrak{T} \otimes \mathfrak{T} \otimes \mathfrak{T}$ is $\mathfrak{K} \otimes \mathfrak{K} \otimes \mathfrak{K}$, and it is contained in every nontrivial ideal of the algebra.
• The result generalizes to higher-rank domains by using analogous symbol map decompositions to describe the ideal structure.
• The analysis confirms that no $C^*$-algebra homomorphism exists from the Toeplitz algebra of the full Hardy module to that of a quotient module, due to non-commutativity of the quotient algebra.