In case $A$ is a symmetric finite dimensional algebra and $e$ an idempotent, $eAe$ is again symmetric.

Is there an easy counterexample for the following:

In case $A$ is additionally a periodic algebra, $eAe$ is also periodic?

Is there a criterion when $eAe$ is still periodic depending on $A$ and $e$?

In case $A$ is a symmetric finite dimensional algebra and $e$ an idempotent, $eAe$ is again symmetric.

Is there an easy counterexample for the following:

In case $A$ is additionally a periodic algebra, $eAe$ is also periodic?

Is there a criterion when $eAe$ is still periodic depending on $A$ and $e$?

In case $A$ is representation-finite the claim is true, but I am not very experienced with representation-infinite periodic algebras.

I did some examples with the computer and in those examples the following thing happened: Let $A$ be a symmetric algebra and $W$ the direct sum of all indecomposable projective $A$-modules $P$ such that $soc(P)$ is periodic. Then $End_A(W)$ was a periodic algebra.