‎We associate a Banach algebra $\\\\\\\\\\\\\\\\A_\\\\\\\\\\\\\\\\Phi$ to each bounded bilinear map $\\\\\\\\\\\\\\\\Phi‎: ‎\\\\\\\\\\\\\\\\X\\\\\\\\\\\\\\\\times \\\\\\\\\\\\\\\\Y\\\\\\\\\\\\\\\\rightarrow \\\\\\\\\\\\\\\\Z$ on Banach spaces‎, ‎and we find out that this Banach algebra can be useful for many purposes in the theory of Banach algebras‎. ‎For example‎, ‎the construction of $\\\\\\\\\\\\\\\\A_\\\\\\\\\\\\\\\\Phi$ enables us to provide many simple examples of Banach algebras with different topological centers‎, ‎which are neither Arens‎ ‎regular nor either left or right strongly Arens irregular‎. ‎It also gives examples of Banach algebras which are not $n-$weakly amenable for each natural number $n.$ We also find out that the dual of $\\\\\\\\\\\\\\\\A_\\\\\\\\\\\\\\\\Phi$ does not enjoy the factorization property of any level‎.

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