Toeplitz Matrices acting on the $\ell^2$-space of an imprimitivity bimodule

TL;DR

This paper defines Toeplitz matrices on the $\, ext{l}^2$-space of an imprimitivity bimodule over a $C^*$-algebra and characterizes them via the crossed product construction.

Contribution

It introduces a new framework for Toeplitz matrices acting on bimodule spaces and characterizes their structure through crossed product representations.

Findings

Toeplitz matrices are characterized as closures of the left regular representation images.

Provides a new perspective on Toeplitz operators in the context of $C^*$-algebras and bimodules.

Establishes a link between Toeplitz matrices and crossed product $C^*$-algebras.

Abstract

We give a definition of Toeplitz matrix acting on the $\ell^2$-space of an imprimitivity bimodule $X$ over a $C^*$-algebra $A$. We characterize the set of Toeplitz matrices as the closure in a certain topology of the image of the left regular representation of the crossed product $A\rtimes X$.