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We introduced the concept of strong property $\mathbb{B}$ with a constant for Banach algebras and, by applying certain analysis on the Fourier algebra of a unit circle, we show that all $C^*$-algebras and group algebras have the strong property $\mathbb{B}$ with a constant given by $288\pi(1+\sqrt{2})$. We then use this result to find a concrete upper bound for the hyperreflexivity constant of certain spaces of bounded $n$-cocycles from $A$ into $X$, where $A$ is a $C^*$-algebra or the group algebra of a group with an open subgroup of polynomial growth and $X$ is a Banach $A$-bimodule. As another application, we show that for a locally compact amenable group $G$ and $1< p< \infty $, the space $CV_P(G)$ of convolution operators on $L^p(G)$ are hyperreflexive with a constant given by $288\pi(1+\sqrt{2})$. This is the generalization of a well-known result of E. Christiensen for $p=2$. This is a joint work with Jafar Soltani Farsani.