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The famous "unit conjecture" for group rings states that all units of a group ring $K[G]$ are trivial for a field $K$ and a torison-free group $G$. We are far away from solving the conjecture (See e.g. http://people.maths.ox.ac.uk/craven/docs/seminars/220909slides.pdf).

I wonder whether more is known about unitary units: Given a group homomorphism $w\colon G \rightarrow \mathbb{Z}/2\mathbb{Z}=\left\{\pm 1\right\}$ we can define an involution on $K[G]$ via $g\in G \mapsto \overline{g}=w(g)g^{-1}$ . The unitary units are those elements $x$ of the group ring with $\overline{x}x=1$. Do we know they are trivial for torsion-free groups?

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    $\begingroup$ In your definition it would be better to replace $\mathbb{Z}/2\mathbb{Z}$ by the subgroup $\{1,−1\}$ of $(K^\times, .)$. These involutions were introduced by Novikov. To the best of my acknowledge, there are no results concerning a unitary version of the unit conjecture. $\endgroup$
    – Salvatore Siciliano
    Commented Nov 1, 2011 at 14:24
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    $\begingroup$ @Joerg: Thank for for posting the link to the very interesting slides! If you haven't already seen it, you may find my related post from yesterday interesting: mathoverflow.net/questions/79559/… $\endgroup$
    – Johan Öinert
    Commented Nov 1, 2011 at 15:15
  • $\begingroup$ Let's make it more specific: do we have any idea about the unitary unit or unit conjecture for the finite fields $K=\mathbb{Z}/p\mathbb{Z}$? $\endgroup$
    – Joerg Sixt
    Commented Nov 2, 2011 at 9:06

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