Gel'fand Theory of the Commutative Banach Algebra $\mathcal A \times_{c} \mathcal I$ with the Convolution Product

Abstract

Let $\mathcal A$ be an algebra and $\mathcal I$ be an ideal in $\mathcal A$. Then $\mathcal A \times \mathcal I$ is an algebra with pointwise linear operations and the convolution product $(a, x) (b, y) = (ab+xy, ay+xb) \; ((a, x), (b, y) \in \mathcal A \times \mathcal I)$; it will be denoted by $\mathcal A \times_{c} \mathcal I$. If $\mathcal A$ is a commutative Banach algebra and $\mathcal I$ is a closed ideal in $\mathcal A$, then $\mathcal A \times_{c} \mathcal I$ is also a commutative Banach algebra with some suitable norm. In this paper, we shall study the Gel'fand theory, uniqueness properties, and regularity of $\mathcal A \times_{c} \mathcal I$.