Trace methods in twisted group algebras (Q2701608)

Let \(K^tG\) be a twisted group algebra of the multiplicative group \(G\) over a field \(K\) of characteristic \(p\geq 0\) with a twisted function \(t\colon G\times G\to K^*\) and a identity element \(\overline 1\) (\(1\in G\)). For every element \(\alpha\in K^tG\) let \(\text{tr }\alpha\) be the coefficient of \(\overline 1\) in \(\alpha\). It is well known [\textit{A. E. Zalesskij}, Dokl. Akad. Nauk SSSR 203, 749-751 (1972; Zbl 0257.16010)] that if \(e\) is an idempotent of a group algebra \(KG\), then \(\text{tr }e\) is an element of the prime subfield of \(K\). Furthermore, if \(p>0\) and \(\alpha\) is a nilpotent element of \(K^tG\) such that its support contains no \(p\)-elements, then \(\text{tr }\alpha=0\) [\textit{D. S. Passman}, Proc. Lond. Math. Soc., III. Ser. 20, 409-437 (1970; Zbl 0192.35802)]. If \(p=0\), then this fact was proved by \textit{J. M. Dimitrova} [Subrings and radicals of crossed products, Ph. thesis, Sofia (1999)] when \(t(H,H)\) is contained in a finitely generated subgroup of \(K^*\) for every finitely generated subgroup \(H\) of \(G\). In the present paper in a very nice manner the author proves that this restriction is not necessary. Moreover, he enlarges the result of Zalesskij for the idempotents of \(K^tG\) and proves some facts for \(\text{tr }\alpha\) of the algebraic elements \(\alpha\in K^tG\).