We introduce filling families with matrix diagonalization as a refinement of the work by R{o}rdam and the first named author. As an application we improve a result on local pure infiniteness and show that the minimal tensor product of a strongly pure
ly infinite $C^*$-algebra and a exact $C^*$-algebra is again strongly purely infinite. Our results also yield a sufficient criterion for the strong pure infiniteness of crossed products $Artimes_varphi mathbb{N}$ by an endomorphism $varphi$ of $A$ (cf. Theorem 7.6). Our work confirms that the special class of nuclear Cuntz-Pimsner algebras constructed by Harnisch and the first named author consist of strongly purely infinite $C^*$-algebras, and thus absorb $mathcal{O}_infty$ tensorially.
The only C*-algebras that admit elimination of quantifiers in continuous logic are $mathbb{C}, mathbb{C}^2$, $C($Cantor space$)$ and $M_2(mathbb{C})$. We also prove that the theory of C*-algebras does not have model companion and show that the theory
of $M_n(mathcal {O_{n+1}})$ is not $forallexists$-axiomatizable for any $ngeq 2$.
