Call a commutative Banach algebra A a $gamma$-algebra if it contains a bounded group $Gamma$ such that $overline{aco(Gamma)}$ contains a multiple of the unit ball of A. In this paper, first by exhibiting several concrete examples, we show that the class of $gamma$-algebras is quite rich. Then, for a $gamma$-algebra A, we prove that $A^star$ has the Schur property iff the Gelfand spectrum $sum$ of A is scattered if $A^star$=ap(A) iff $A^star=overline{Span(sum)}$.