On certain Toeplitz operators and associated completely positive maps

TL;DR

This paper investigates Toeplitz operators linked to commuting operator tuples influenced by complex geometry, characterizes dual operators, and proves an extension theorem relevant for dilation theory in multiple variables.

Contribution

It introduces a new framework for Toeplitz operators associated with complex geometric conditions and establishes an extension theorem that addresses limitations of dilation theorems for higher dimensions.

Findings

Toeplitz algebra is homeomorphic to L^fty of a compact subset of c^n

Characterization of dual Toeplitz operators

Proved an extension theorem relevant for dilation theory in multiple variables

Abstract

We study Toeplitz operators with respect to a commuting $n$-tuple of bounded operators which satisfies some additional conditions coming from complex geometry. Then we consider a particular such tuple on a function space. The algebra of Toeplitz operators with respect to that particular tuple becomes naturally homeomorphic to $L^\infty$ of a certain compact subset of $\mathbb C^n$. Dual Toeplitz operators are characterized. En route, we prove an extension type theorem which is not only important for studying Toeplitz operators, but also has an independent interest because dilation theorems do not hold in general for $n>2$.