Spectral properties and stability of perturbed Cartesian product (Q1680375)

Given two Banach algebras \(\mathcal A\) and \(\mathcal B\) and a homomorphism \(T: \mathcal A \to \mathcal B\) with \( \| T\| \leq 1\), a new Banach algebra structure is defined on the Cartesian product \(\mathcal A \times \mathcal B\) by \[ (a,b) (c,d)= (ac+a Td+ (Tb)c, bd), \quad \|(a,b)\|= \| a\|+ \| b\|,\quad a \in A, \quad b \in B. \] This algebra is denoted by \(\mathcal A \times_T \mathcal B\), and is seen as a perturbation of the usual Cartesian product (see [\textit{S. J. Bhatt} and \textit{P. A. Dabhi}, Bull. Aust. Math. Soc. 87, No. 2, 195--206 (2013; Zbl 1282.46041)]. The main goal of the present paper is to investigate the stability of many spectral properties in the commutative case.