We introduce algebras which are inductive limits of Banach spaces and carry inequalities which are counterparts of the inequality for the norm in a Banach algebra. We then define an associated Wiener algebra, and prove the corresponding version of the well-known Wiener theorem. Finally, we consider factorization theory in these algebra, and in particular, in the associated Wiener algebra.
D. Alpay and G. Salomon. On algebras which are inductive limits of Banach spaces. Integral Equations and Operator Theory, vol. 83, Issue 2 (2015), pp. 211-229.