Automatic Continuity of Almost Multiplicative Maps Between Frechet Algebras

For Fracutemathbftexte acutemathbftexte chet algebras (A,(p n)) (A,(pn)) and (B,(q n)) (B,(qn)), a linear map T:ArightarrowB T:ArightarrowB is textit{almost multiplicative} with respect to (p n) (pn) and (q n) (qn), if there exists varepsilongeq0 varepsilongeq0 such that q n (Tab−TaTb)leqvarepsilonp n (a)p n (b), qn(Tab−TaTb)leqvarepsilonpn(a)pn(b), for all ninmathbbN ninmathbbN, a,binA a,binA, and it is called textit{weakly almost multiplicative} with respect to (p n) (pn) and (q n) (qn), if there exists varepsilongeq0 varepsilongeq0 such that for every kinmathbbN kinmathbbN, there exists n(k)inmathbbN n(k)inmathbbN, satisfying the inequality q k (Tab−TaTb)leqvarepsilonp n(k) (a)p n(k) (b), qk(Tab−TaTb)leqvarepsilonpn(k)(a)pn(k)(b), for all a,binA a,binA. We investigate the automatic continuity of (weakly) almost multiplicative maps between certain classes of Fracutemathbftexte acutemathbftexte chet algebras, such as Banach algebras and Fracutemathbftexte acutemathbftexte chet Q Q -algebras. We also obtain some results on the automatic continuity of dense range (weakly) almost multiplicative maps between Fracutemathbftexte acutemathbftexte chet algebras.