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We also introduce some new concepts as left-weak$^*$-weak convergence property [$Lw^*wc-$property] and right-weak$^*$-weak convergence property [$Rw^*wc-$property] and for Banach algebra $A$, suppose that $A^*$ and $A^{**}$, respectively, have $Rw^*wc-$property and $Lw^*wc-$property. 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