Mathematics – Functional Analysis
Scientific paper
1999-08-15
Mathematics
Functional Analysis
44 pages
Scientific paper
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.
No associations
LandOfFree
Uncountable direct systems and a characterization of non-separable projective $C^{\ast}$-algebras does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Uncountable direct systems and a characterization of non-separable projective $C^{\ast}$-algebras, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Uncountable direct systems and a characterization of non-separable projective $C^{\ast}$-algebras will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-388901