1. ## Banach Algebra

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 under the maximum norm?

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?

2. We discourage double-posting on this forum.

3. I understand that but I was worry that I post the my thread in the wrong topic so I posted it again here.

4. Other Advanced Topics is just fine. Odds are that someone competent to help you (I'm not, incidentally) will be monitoring Other Advanced Topics anyway. Follow the rules (now posted as a sticky in every subforum), and you'll do better!