# Math Help - Banach Algebra

1. ## Banach Algebra

Let $A$ be a Banach Algebra without unit and be embeddd into a unital Banach Algebra $A^+$such that the elements in $A^+$are of the form $(a,\alpha),$ $a \in A$ and $\alpha \in \mathbb{C}.$

Define the multiplication in $A^+$ by $(a,\alpha)(b,\beta)=(ab+\alpha b + \beta a,\alpha \beta)$ and the involution by $(a,\alpha)^*=(a^*,\alpha^-)$where $\alpha^-$ is the conjugate of $\alpha$.

It is well known that under the norm $\|(a,\alpha)\|=\|a\|+|\alpha|, A^+$ is a Banach algebra. However, if the norm is defined as $\|(a,\alpha)\|=max\{\|a\|,|\alpha|\}$, is $A^+$still a Banach Algebra?

I try to prove that $\|(a,\alpha)(b,\beta)\| \le \|(a,\alpha)\|\|(b,\beta)\|$ does not hold for a specific element in $A^+.$However, I still do not get a correct one. Can anyone help?