An example of a liminal $C^{\ast }$-algebra

Abstract:

For each countable ordinal $\gamma$ there exists a unital separable liminal ${C^ \ast }$-algebra ${A_\gamma }$ with the property that if $({I_\rho })_{\rho = 1}^\beta$ is any composition sequence of ${A_\gamma }$ such that the spectra of the quotients ${I_{\rho + 1}}/{I_\rho }$ are Hausdorff, then $\beta \geqslant \gamma + 1$. Moreover, there is a composition sequence $({I_\rho })_{\rho = 1}^{\gamma + 1}$ of ${A_\gamma }$ such that the spectra of the quotients ${I_{\rho + 1}}/{I_\rho }$ are Hausdorff.