Identities on units of algebraic algebras
Let \(f(\zeta_1,\zeta_2,\dots,\zeta_n)\) be a Laurent polynomial over the field \(K\) so that \(f\in K[\zeta_1,\zeta_1^{-1},\dots,\zeta_n,\zeta_n^{-1}]\). If \(A\) is a \(K\)-algebra with group of units \(U(A)\), then it makes sense to consider evaluations of the form \(f(u_1,u_2,\dots,u_n)\) with \(u_1,u_2,\dots,u_n\in U(A)\), and then \(U(A)\) is said to satisfy a Laurent polynomial identity (LPI) if there exists a nonzero \(f\) as above with \(f(u_1,u_2,\dots,u_n)=0\) for all \(u_i\in U(A)\). It is shown here that if \(A\) is an algebraic algebra over an infinite field \(K\) and if \(U(A)\) satisfies an LPI, then \(A\) satisfies a polynomial identity. Furthermore, if \(A\) is noncommutative and if the noncentral units of \(A\) satisfy an LPI, then again \(A\) satisfies a polynomial identity.
scientific article; zbMATH DE number 1763549