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:



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



Example to periodic symmetric algebras
The simplest counterexample I know is when A is the mesh algebra of generalized Dynkin type $C\_3$ (see the paper "Periodic algebras" by Erdmann and Skowronski). Its quiver is the same as the preprojective algebra of Dynkin type $D\_4$, but the relations are slightly different. It is symmetric and periodic of period $6$. If the central vertex is labeled 2, and the three other vertices are 0, 1, and 3, the indecomposable projective $e\_2A$ has structure:
$$\begin{array}{ccc} & 2 \\ 0 & 1 & 3 \\ 2 & \ 2 \\ 0 & 1 & 3 \\ & 2\end{array}$$
Thus $e\_2Ae\_2 \cong k[x,y]/(xy, x^2+y^2)$, which is not periodic.


Your other observation at the end is interesting. While I don't expect that to hold in general, at the moment I don't know a counterexample.


