Functoriality of real crossed product K-theory spectral sequences with respect to group homomorphisms

Abstract

Spectral sequences are a key tool for computing the K-theory of a crossed product C${}^{\ast }$-algebra. However, the impact of a group homomorphism $\Omega :G\to H$ on such a spectral sequence was unknown until quite recently, even when $G={\mathbb{Z}}^{\ell }$, $H={\mathbb{Z}}^k.$ Recent work [Mil25] of the fourth-named author in the complex case establishes that ABC spectral sequences are functorial with respect to group homomorphisms. In this paper, we obtain the analogous result for real K-theory and for united K-theory. Specifically, we first show that the ABC spectral sequence approximates KO${}_{\ast }(G{⋉}_{r}A)$ with the group homology H${}_p(G;K{O}_q(A))$ when $G$ is a torsion-free discrete group satisfying the Baum--Connes conjecture with coefficients in $A$. Then, for a homomorphism $\Omega :G\to H$ of such groups with amenable kernel, and a real $H$-C${}^{\ast }$-algebra $A$, we show moreover that the map in K-theory induced by the $\ast$-homomorphism $G{⋉}_{r}A\to H{⋉}_{r}A$ is approximated by the natural map in group homology.