Let $A$ be a non-unital Banach algebra, and let $A^+$ be its unitization. Then $||(a,z)||_1=||a||+|z|$ is a Banach algebra norm on $A^+$.
Can we also make $A^+$ a Banach algebra by giving it the norm $||(a,z)||_p=(||a||^p+|z|^p)^{1/p}$ for $p\in(1,\infty)$?
Remark: My original post was also about $||(a,z)||_{op}=sup\{||ab+zb||:b\in A,||b||\leq 1\}$ but then I realized that it may not even be a norm in general.
No, not in general. For instance, suppose $A=\mathbb{C}$ with the standard norm. Then (via a change of basis $(a,z)\mapsto (a+z,z)$) we can identify $A_+$ with $\mathbb{C}^2$ with coordinatewise multiplication, and your norm with $\|(a,b)\|=(|a-b|^p+|b|^p)^{1/p}$. Now consider the elements $x=(1,0)$ and $y=(1,1/2)$. We have $\|x\|=1$ and $\|y\|=2^{(1-p)/p}<1$, but $\|xy\|=\|x\|=1>\|x\|\|y\|$.