Two generalizations of the Gleason-Kahane-Żelazko theorem (Q1816510)

We consider a unital Banach algebra \({\mathfrak A}\), and a continuous unital linear mapping \(\varphi\) of \({\mathfrak A}\) into \(M_n (\mathbb{C})\) -- the \(n \times n\) matrices over \(\mathbb{C}\). The first generalization states that if \(\varphi\) sends invertible elements to invertible elements, then the kernel of \(\varphi\) is contained in a proper two sided closed ideal of finite codimension. The second result characterizes this property for \(\varphi\) in saying that \(\varphi ({\mathfrak A}_{\text{inv}})\) is contained in \(\text{GL}_n (\mathbb{C})\) if and only if for each \(a\) in \({\mathfrak A}\) and each natural number \(k\): \[ \text{trace} (\varphi (a^k))= \text{trace} (\varphi (a)^k). \]