Algebras whose units satisfy a ∗-Laurent polynomial identity

Let R be an algebraic algebra over an infinite field and ∗ be an involution on R . We show that if the units of R , U ( R ) , satisfy a ∗ -Laurent polynomial identity, then R satisfies a polynomial identity. Also, let G be a torsion group and F a field. As a generalization of Hartley’s Conjecture, in Broche et al. (Arch Math 111:353–367, 2018), it is shown that if U ( F G ) satisfies a Laurent polynomial identity which is not satisfied by the units of the relative free algebra F [ α , β : α 2 = β 2 = 0 ] , then FG satisfies a polynomial identity. In this paper, we instead consider non-torsion groups G and provide some necessary conditions for U ( F G ) to satisfy a Laurent polynomial identity.