Minkowski type inequality in Banach algebras

Question

Under which circumstances it is true that $\|(A+B)^n\|^{1/n}\le \|A^n\|^{1/n}+\|B^n\|^{1/n}$ for elements $A$ and $B$ in a Banach algebra and a natural number $n$?

Answer 1

The case $n=1$ holds always.

For $n\geq2$, i don't have any kind of general answer, but I expect such inequality to hold basically for norms that behave like the one-norm, and little else.

As an example, the inequality fails for any norm in the $2\times2$ matrices. Indeed, if $$ A=\begin{bmatrix}0&1\\0&0\end{bmatrix},\ \ B=\begin{bmatrix}0&0\\1&0\end{bmatrix}, $$ then $$ \|(A+B)^n\|>0, \ \mbox{ while } \ A^n=B^n=0. $$ And of course this idea works for any algebra where you have nilpotent elements with non-nilpotent sum.