Algebraic Elements in Group Rings

In this brief note, we study algebraic elements in the complex group algebra C[G]. Specifically, suppose £ e C[G] satisfies /(£) = 0 for some nonzero polynomial f(x) e C[x]. Then we show that a certain fairly natural function of the coefficients of Z is bounded in terms of the complex roots of f(x). For G finite, this is a recent observation of [HLP], Thus the main thrust here concerns infinite groups, where the inequality generalizes results of [K] and [W] on traces of idempotents. Introduction Let a = J2geK a g belong to the group algebra F[G]. If k is a conjugacy class of 67, then the k-trace of a is defined by aK = ¿^,g€K a . It is clear that the map : F[G] —► F is F -linear. Furthermore, if g,h e G, then hg = g~\gh)g so (gh hg)K = 0. Thus, by linearity, (aß)K = (ßa)K for all a, ß € F[G] and K is indeed a trace map. In this paper we study algebraic elements in the complex group algebra C[C7] and our goal is to prove Theorem 1. Let £ be an element of C[G] and suppose that f(c¡) = 0 for some nonzero polynomial f(x) e C[x]. If X denotes the maximum of the absolute values of the complex roots of f, then E<^/w^2 K where \k\ is the size of the class and ~ denotes complex conjugation. For G finite, this is a result of [HLP] which was proved using character theory. So the real content here concerns infinite groups. In this case, if k is a conjugacy class of infinite size, then the summand ÇkÇk/\k\ is obviously zero and hence has no effect on the above formula. Thus we need only restrict our attention to the conjugacy classes of G of finite size. Two special cases of the theorem are worth mentioning. First, if c¡ is nilpotent, then X = 0 and hence we have ÇK = 0 for all finite classes k . Second, Received by the editors May 4, 1989 and, in revised form, June 27, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 16A27. The first author is a visiting professor, University of California, Los Angeles, California 90024. Research of the second author was supported in part by NSF Grant No. DMS 8521704. ©1990 American Mathematical Society 0002-9939/90 $1.00+ $.25 per page