On crossed product rings with twisted involutions, their module categories and L-theory

We study the Farrell-Jones Conjecture with coefficients in an additive G-category with involution. This is a variant of the L-theoretic Farrell-Jones Conjecture which originally deals with group rings with the standard involution. We show that this formulation of the conjecture can be applied to crossed product rings R*G equipped with twisted involutions and automatically implies the a priori more general fibered version.

💡 Research Summary

The paper addresses a significant extension of the L‑theoretic Farrell‑Jones Conjecture (FJC) to a class of non‑commutative rings equipped with non‑standard, “twisted” involutions. Classical formulations of the conjecture deal with group rings ℤG or RG (R a regular ring) together with the canonical involution that sends a group element g to its inverse and applies the standard involution on the coefficient ring. However, many natural algebraic objects—most notably crossed product rings R∗G—carry additional 2‑cocycle data τ : G×G→U(R) and 1‑cocycle data c : G→U(R) which modify both the multiplication and the involution. The authors call the resulting involution “twisted” because it incorporates these cocycles in a non‑trivial way.

The first technical contribution is a precise definition of the twisted involution on a crossed product R∗G. Given an involution * on R and the group inversion g↦g⁻¹, the involution on the crossed product is defined by
\