Let \(\Gamma\) be a discrete group and \(R\) a unital ring. Using continuously controlled algebra, one can define a functor \(F\), from the category of \(\Gamma{-}\)CW complexes to the category of spectra, such that \(\pi_*(F(-))\) satisfies the Eilenberg-Steenrod axioms, except for the dimension axiom, and \(\pi_*(F(\bullet))=K_*(R\Gamma)\) [I. Hambleton and E. K. Pedersen, Math. Ann. 328, 27-57 (2004; Zbl 1051.19002)]. Let \(\mathcal E=E\Gamma(\mathfrak f)\) denote the universal space for \(\Gamma\)-actions with finite isotropy, where \(\mathfrak f\) denotes the family of finite subgroups of \(\Gamma\). The main result of the paper is split injectivity of the assembly map \(\pi_*(F(\mathcal E))\to\pi_*(F(\bullet))\) provided that \(\mathcal E\) satisfies certain geometric conditions.