A universal coefficient theorem for twisted K-theory

In this paper, we recall the definition of twisted K-theory in various settings. We prove that for a twist $\tau$ corresponding to a three dimensional integral cohomology class of a space X, there exist a “universal coefficient” isomorphism K_{}^{\tau}(X)\cong K_{}(P_{\tau})\otimes_{K_{}($\mathbb{C}$P^{\infty})} \hat{K}_{} where $P_{\tau }$ is the total space of the principal $\mathbb{C}{P}^{\mathrm{\infty }}$-bundle induced over X by $\tau$ and ${\widehat{K}}_{\ast }$ is obtained form the action of $\mathbb{C}{P}^{\mathrm{\infty }}$ on K-theory.