Various Spectral Properties in the Banach Algebra $\mathcal A \times_{c} \mathcal I$ with the Convolution Product

Abstract

Let $\mathcal I$ be an ideal of an associative algebra $\mathcal A$ over the field $\mathbb C$. Then the vector space $\mathcal A \times \mathcal I$ with pointwise linear operations becomes an algebra with the product $(a, x) (b, y) = (ab+xy, ay+bx) \; ((a, x), (b, y) \in \mathcal A \times \mathcal I).$ This product is known as the convolution product and this algebra is denoted by $\mathcal A \times_{c} \mathcal I$. In this paper, some well-known spectral properties of Banach algebras are studied for $\mathcal A \times_{c} \mathcal I$.