We introduce the concept of a direct $C_{ \omega}^{ \ast}$-system and show that every non-separable unital $C^{ \ast}$-algebra is the limit of essentially unique direct $C_{ \omega}^{ \ast}$-system. This result is then applied to the problem of characterization of projective unital $C^{ \ast}$-algebras. It is shown that a non-separable unital $C^{ \ast}$-algebra $X$ of density $\tau$ is projective if and only if it is the limit of a well ordered direct system ${ \mathcal S}_{ X} = \{ X_{ \alpha}, i_{ \alpha}^{ \alpha +1}, \alpha < \tau \}$ of length $\tau$, consisting of unital projective $C^{ \ast}$-subalgebras $X_{ \alpha}$ of $X$ and doubly projective homomorphisms (inclusions) $i_{ \alpha}^{ \alpha +1} \colon X_{ \alpha} \to X_{ \alpha +1}$, $\alpha < \tau$, so that $X_{ 0}$ is separable and each $i_{ \alpha}^{ \alpha +1}$, $\alpha < \tau$, has a separable type. In addition we show that a doubly projective homomorphism $f \colon X \to Y$ of unital projective $C^{ \ast}$-algebras has a separable type if and only if there exists a pushout diagram \[ \begin{ CD} X @>f>> Y @A{ p}AA @AA{ q}A X_{ 0} @>f_{ 0}>> Y_{ 0}, \end{ CD} \] \noindent where $X_{ 0}$ and $Y_{ 0}$ are separable unital projective $C^{ \ast}$-algebras and the homomorphisms $i_{ 0} \colon X_{ 0} \to Y_{ 0}$, $p \colon X_{ 0} \to X$ and $q \colon Y_{ 0} \to Y$ are doubly projective. These two results provide a complete characterization of non-separable projective unital $C^{ \ast}$-algebras in terms of separable ones.
Categories: math.FA