ALGEBRAIC ALGEBRAS WITH INVOLUTION susan montgomery

The following theorem is proved: Let R be an algebra with involution over an uncountable field F. Then if the symmetric elements of R are algebraic, R is algebraic. In this paper we consider the following question: "Let R be an algebra with involution over a field F, and assume that the symmetric elements S of R are algebraic over F. Is R algebraic over FT* Previous results related to this question have been obtained by restricting the kind of algebraic relationships satisfied by the symmetric elements. For example, it was shown by Baxter and Martindale [1] for fields of characteristic not 2, and later by the author [5] for arbitrary fields, that if the symmetric elements are algebraic of bounded degree (or more generally, satisfy a polynomial identity), then R must be algebraic. Another such result concerns rings whose symmetric elements are periodic (that is, for each seS, there is some integer n(s)> 1 such that sn{s)=s). In this case, the author has shown [6], [7] that R must be algebraic; in fact R satisfies a polynomial identity. When R is a division ring, much more can be said: I. N. Herstein and the author [2] have shown that R must actually be commutative. Finally, it has been shown by Osborn [8] that if S is nil and F is uncountable, then R is nil. This answers for uncountable fields a question of McCrimmon [4, p. 391]: "If Sis nil, is R nil?" An affirmative answer to this question in general would follow from an affirmative answer to the first question. For, as has been observed by both McCrimmon [4, p. 390] and Osborn [8, p. 306], if Sis nil then R must be a radical ring. But if R is algebraic, every element of the radical is nil; thus R would be nil. The result presented here differs from those described above in that no additional restrictions are imposed on the symmetric elements. We prove: Theorem. Let R be an algebra with involution over an uncountable field F. Then if the symmetric elements of R are algebraic, R is algebraic. Received by the editors January 8, 1971. AMS 1969 subject classifications. Primary 1658.