An algebra of distributions on an open interval

Let $(a,b)$ be any open subinterval of the reals which contains the origin and let $\mathfrak {B}$ denote the family of all distributions on $(a,b)$ which are regular in some interval $( \in ,0)$, where $\in < 0$. Then $\mathfrak {B}$ is a commutative algebra: Multiplication is defined so that, when restricted to those distributions on $(a,b)$ whose supports are contained in $[0,b)$, it is ordinary convolution. Also, $\mathfrak {B}$ can be injected into an algebra of operators; this family of operators is a sequentially complete locally convex space. Since it preserves multiplication, this injection serves as a generalization (there are no growth restrictions) of the two-sided Laplace transformation.