Let $\displaystyle A$ be a Banach Algebra without unit and be embeddd into a unital Banach Algebra $\displaystyle A^+$such that the elements in $\displaystyle A^+$are of the form $\displaystyle (a,\alpha),$$\displaystyle a \in A$ and $\displaystyle \alpha \in \mathbb{C}.$
Define the multiplication in $\displaystyle A^+$ by $\displaystyle (a,\alpha)(b,\beta)=(ab+\alpha b + \beta a,\alpha \beta)$ and the involution by $\displaystyle (a,\alpha)^*=(a^*,\alpha^-)$where $\displaystyle \alpha^-$ is the conjugate of $\displaystyle \alpha$.
It is well known that under the norm $\displaystyle \|(a,\alpha)\|=\|a\|+|\alpha|, A^+$ is a Banach algebra. However, if the norm is defined as $\displaystyle \|(a,\alpha)\|=max\{\|a\|,|\alpha|\}$, is $\displaystyle A^+$still a Banach Algebra?
I try to prove that$\displaystyle \|(a,\alpha)(b,\beta)\| \le \|(a,\alpha)\|\|(b,\beta)\|$ does not hold for a specific element in $\displaystyle A^+.$However, I still do not get a correct one. Can anyone help?