Two generalizations of the Gleason-Kahane-Z̀elazko theorem

In this article we obtain 2 generalizations of the well known Gleason-Kahane-Zelazko Theorem. We consider a unital Banach algebra 21, and a continuous unital linear mapping φ of 21 into M_n(ℂ) - the n x n matrices over ℂ. The first generalization states that if φ sends invertible elements to invertible elements, then the kernel of φ is contained in a proper two sided closed ideal of finite codimension. The second result characterizes this property for φ in saying that φ(21_inv) is contained in GL_n(ℂ) if and only if for each o in 21 and each natural number k: trace(φ(a^k)) = trace(φ(a)^k).