Commutativity theorems and projection on center of a Banach algebra

Abstract

Let $ \mathcal{X}$ be a Banach algebra. In this article, on the one hand,  we proved  some results concerning the  continuous projection from $ \mathcal{X}$ to its center. On the other hand, we investigate the commutativity of $\mathcal{X}$ under specific conditions. Finally, we included some examples and applications to prove that various restrictions in the hypotheses of our theorems are necessary.