Toeplitz algebras and the Heisenberg group

TL;DR

This paper establishes an isomorphism between a Toeplitz algebra constructed from d-shifts and an ideal of the C*-algebra of a (2d+1)-dimensional Heisenberg group, illustrating a specific case of a broader theoretical framework.

Contribution

It provides a simple proof of a special case of a general result relating Toeplitz algebras and nilpotent Lie groups, emphasizing basic functional analysis techniques.

Findings

Demonstrates an explicit isomorphism between Toeplitz algebra and Heisenberg group C*-algebra ideal.

Connects Toeplitz algebra structures with nilpotent Lie group C*-algebras.

Simplifies the proof of a general result using elementary functional analysis.

Abstract

We show an isomorphism between an algebra which is naturally constructed from the Toeplitz algebra generated by d-shifts, and an ideal of the C * -algebra of the (2d + 1)-dimensional Heisenberg group. This is a particular case of a more general result for graded nilpotent Lie groups involving symbols in the filtered calculus. The proof presented here however only involves basic functional analysis while still showcasing the ideas of the proof in the general setting.