First Hachschild Cohomology Group of Triangular Banach Algebras on Induced Semigroup Algebras

Let S be a discrete semigroup with a left multiplier operator T on S. A new product on S de ned by T related to S and T creates a new induced semigroup ST . Suppose that T is bijective and T1 = [ .1(S) .1(S) .1(S) ] and T2 = [ .1(ST ) .1(ST ) .1(ST ) ] : In this paper, we show that the rst cohomology groups H1(T1; T 1 ) and H1(T2; T 2 ) are equal. Therefore T1 is weakly amenable if and only if T2 is weakly amenable.