Simple Jordan Color Algebras Arising from Associative Graded Algebras

If R is a G-graded associative algebra, where G is an abelian group and ε is a bicharacter for G, then R also has the structure of a Jordan color algebra. In addition, if R is endowed with a color involution * , then the symmetric elements S under * are also a Jordan color algebra. Generalizing results of I. N. Herstein, we examine the Jordan color structure of R and S. In particular, we show that if R is a graded-simple algebra, then both R and S are simple Jordan color algebras, except for some special cases which cannot arise in the ordinary case.