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, $${\mathcal {U}}(R)$$ , 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 $${\mathcal {U}}(FG)$$ satisfies a Laurent polynomial identity which is not satisfied by the units of the relative free algebra $$F[\alpha , \beta :\alpha ^2=\beta ^2=0]$$ , then FG satisfies a polynomial identity. In this paper, we instead consider non-torsion groups G and provide some necessary conditions for $${\mathcal {U}}(FG)$$ to satisfy a Laurent polynomial identity.