On graded polynomial identities with an antiautomorphism (Q1858234)

Let \(G\) be a monoid with cancellation. In a recent paper [Commun. Algebra 31, No. 6, 2951-2964 (2003; Zbl 1035.16014)] \textit{K. I. Beidar} and \textit{M. A. Chebotar} found various conditions under which an almost \(G\)-graded algebra satisfying a \(G\)-graded polynomial identity is a PI algebra. In the paper under review the authors extend these results to graded polynomial identities with an antiautomorphism \(\phi\). Several conditions implying that a \(G\)-graded algebra with \(\phi\) is PI are presented, and their necessity is justified by different examples. We state here only a simplified version of one of the results: Let \(G\) be commutative and let \(R\) be a strongly \(G\)-graded algebra with \(\phi\) having finite \(G\)-grading. If \(R\) satisfies a graded polynomial identity with \(\phi\), then \(R\) is a PI algebra. In particular, the results in this paper generalize \textit{S. A. Amitsur}'s classical theorem [Isr. J. Math. 7, 63-68 (1969; Zbl 0179.33701)] stating that a ring satisfying a polynomial identity with involution is PI.