Group identities on symmetric units under oriented involutions in group algebras

Let \(\mathbb{F}G\) denote the group algebra of a locally finite group \(G\) over the infinite field \(\mathbb{F}\) with \(char(\mathbb{F})\neq 2\), and let \(\circledast:\mathbb{F}G\rightarrow \mathbb{F}G\) denote the involution defined by \(\alpha=\Sigma\alpha_{g}g \mapsto \alpha^\circledast=\Sigma\alpha_{g}\sigma(g)g^{\ast}\), where \(\sigma:G\rightarrow \{\pm1\}\) is a group homomorphism (called an orientation) and \(\ast\) is an involution of the group \(G\). In this paper we prove, under some assumptions, that if the \(\circledast\)-symmetric units of \(\mathbb{F}G\) satisfies a group identity then \(\mathbb{F}G\) satisfies a polynomial identity, i.e., we give an affirmative answer to a Conjecture of B. Hartley in this setting. Moreover, in the case when the prime radical \(\eta(\mathbb{F}G)\) of \(\mathbb{F}G\) is nilpotent we characterize the groups for which the symmetric units \(\mathcal{U}^+(\mathbb{F}G)\) do satisfy a group identity.