If $A$ is a $G$-graded algebra then one can define on it a color involution, i.e. a bijective linear map preserving the grading such that the image of a product of two homogeneous elements is defined through a bicharacter of the group G. More precisely, $*:A\rightarrow A$ is a color involution if
$a^{**}=a,$ for all $a\in A;$
$*$ is linear;
$*(A_g)\subseteq A_g,$ for all homogeneous component $A_g;$
$(ab)^* = \varepsilon(g,h)b^*a^*,$ for all homogeneous elements $a\in A_g$ and $b\in A_h,$ where $\varepsilon$ is a bicharacter of $G.$
Color algebras are strictly related with color Lie and Jordan algebras.
Does exist a classification of simple (associative) algebras with color involution? Is there at least a classification of color involutions on matrix algebras?
