Uncountable direct systems and a characterization of non-separable projective $C^{\ast }$-algebras

Abstract

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) $i_{\alpha }^{\alpha +1}:X_{\alpha }\to X_{\alpha +1}$, $\alpha <\tau$, 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 $f:X\to Y$ 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 $i_0:X_0\to Y_0$, $p:X_0\to X$ and $q:Y_0\to Y$ are doubly projective. These two results provide a complete characterization of non-separable projective unital $C^{\ast }$-algebras in terms of separable ones.

44 pages