application of factorization theorem

Question

Young's inequlity tells us that $L^{1}(\mathbb R)\ast L^{p}(\mathbb R) \subset L^{p}(\mathbb R)$ with norm inequality $$\|f\ast g\|_{L^{p}} \leq \|f\|_{L^1}\|g\|_{L^p};$$

and of course this inequality has lot of importance in Analysis and PDEs. (every body know this, and perhaps all the time we are using these kind of inequality)

On the other hand, Cohen- Hewitt factorization theorem states that,

$L^{p}\subset L^{1}\ast L^{p}, (1\leq p <\infty.)$

My Question are: (i) Can you illustrate some application of this factorization theorem in Analysis and PDEs ? (or some other math branch) (ii) What is an importance of this factorization theorem?

Edit: Some references [paper or books (which contains some application)] will be o.k. for me.

Answer 1

One term to search for is automatic continuity (one major expert is H. Garth Dales who wrote a number of books on the subject, see also the books by Helemskii).

A non-trivial application of the Cohen-Hewitt factorization theorem in this direction is:

Let $A$ be a Banach $\ast$-algebra with bounded approximate unit. Then every positive linear functional $f \colon A \to \mathbb{C}$ is continuous (that is to say, every linear functional satisfying $f(a^\ast a) \geq 0$ for all $a \in A$).

A proof and some further arguments giving an idea of the flavor of the arguments can be found here in part b) of the answer.