Identities on Units of Algebraic Algebras

Let A be an algebraic algebra over an infinite field K and let U(A) be its group of units. We prove a stronger version of Hartley's conjecture for A, namely, if a Laurent polynomial identity (LPI, for short) f=0 is satisfied in U(A), then A satisfies a polynomial identity (PI). We also show that if A is non-commutative, then A is a PI-ring, provided f=0 is satisfied by the non-central units of A. In particular, A is locally finite and, thus, the Kurosh problem has a positive answer for K-algebras whose unit group is LPI. Moreover, f=0 holds in U(A) if and only if the same identity is satisfied in A. The last fact remains true for generalized Laurent polynomial identities, provided that A is locally finite.