Taylor spectrum of a Banach module over the quantum plane

In this talk we discuss the joint spectra of Banach space representations of the quantum q-plane. 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 q-projection property is proved for the Taylor spectrum, which stands out the conventional projection property for a joint spectrum in the commutative case. It is provided the key examples of the Banach q-modules, which do not possess neither forward nor backward projection properties.