Many properties of two-sided algebras remain valid for one-sided algebras. Namely, any one sided Banach algebra is commutative modulo its Jacobson radical.
KINANI, A. EL; NAJMI, A.; and OUDADESS, MOHAMED (2002) "One Sided Banach Algebras," Turkish Journal of Mathematics: Vol. 26: No. 3, Article 6. Available at: https://journals.tubitak.gov.tr/math/vol26/iss3/6