Taylor spectrum of a Banach module over the quantum plane

TL;DR

This paper explores the Taylor spectrum of Banach modules over the quantum plane, introducing joint spectra concepts in a noncommutative setting and establishing properties distinct from classical cases.

Contribution

It defines the joint and Taylor spectra for Banach q-modules, proving a noncommutative projection property and providing key examples lacking traditional projection properties.

Findings

Established the Taylor joint spectrum for Banach q-modules.

Proved the noncommutative projection q-property for the Taylor spectrum.

Provided examples without forward or backward projection properties.

Abstract

In the paper we investigate the joint spectra of Banach space representations of the quantum q-plane called Banach q-modules. Based on the transversality relation from the topological homology of the trivial modules versus given a left Banach q-module, we introduce the joint (essential) spectra of a Banach q-module. In particular, we have the well defined Taylor joint spectrum of a Banach q-module. The noncommutative projection q-property is proved for the Taylor spectrum, which stands out the conventional projection property in the commutative case. It is provided the key examples of the Banach q-modules, which do not possesses nether forward nor backward projection properties.