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Let 
$A=\bigoplus _{i\in \mathbb{Z}}A_{i}$ be a finite-dimensional graded symmetric cellular algebra with a homogeneous symmetrizing trace of degree 
$d$. We prove that if 
$d\neq 0$ then 
$A_{-d}$ contains the Higman ideal 
$H(A)$ and 
$\dim H(A)\leq \dim A_{0}$, and provide a semisimplicity criterion for 
$A$ in terms of the centralizer of 
$A_{0}$.
A ring 
$R$ is called quasi-Baer if the right annihilator of every right ideal of $R$ is generated by an idempotent as a right ideal. We investigate the quasi-Baer property of skew group rings and fixed rings under a finite group action on a semiprime ring and their applications to 
${{C}^{*}}$-algebras. Various examples to illustrate and delimit our results are provided.
