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\(\newcommand{\Cat}{{\rm Cat}} \) \(\newcommand{\A}{\mathcal A} \) \(\newcommand{\freestar}{ \framebox[7pt]{$\star$} }\)

1. Ultrapowers

    1. Problem 1.1.

      Does every separable C*-algebra embed into an ultrapower of $\mathcal O_2$ with respect to an ultrafilter on $\mathbb N$?
        1. Remark. [Kirchberg] Every exact C*-algebra embeds into $\mathcal O_2$.
            •     If $\phi_j\colon A\to \prod_{\mathcal U} \mathcal O_2$ are *-homomorphisms, write $\phi_1\leq \phi_2$ if there is a partial isometry $u$ in $A$ such that $u^*\phi_2 u=\phi_1$.

              Problem 1.2.

              [Kirchberg] Assume $A$ is separable and $A$ embeds into $\prod_{\mathcal U}\mathcal O_2$. Is there a $\leq$-maximal embedding $\phi$ of $A$ into $\prod_{\mathcal U}\mathcal O_2$?
                • Problem 1.3.

                  Can one prove in ZFC that for some free ultrafilter $\mathcal U$ on $\mathbb N$ we have $\mathcal{B}\left( H\right) '\cap \prod_{\mathcal U}\mathcal{B}\left( H\right) = \mathbb C I$?
                      Selective ultrafilters have this property. The Continuum Hypothesis implies the existence of selective ultrafilters.
                    •     An ultrafilter $\mathcal U$ on $\mathbb N$ is flat if there are $h_n\colon \mathbb N\searrow [0,1]$ such that
                      1. $h_n(0)=1$,
                      2. $\lim_j h_n(j)=0$,
                      3. $(\forall f\colon \mathbb N\nearrow \mathbb N)\lim_{n\to \mathcal U} \sup_{j\in \mathbb N} |h_n(j)-h_n(f(j))|=0$.

                      Problem 1.4.

                      Is a nonprincipal ultrafilter such that $\mathcal{B}\left( H\right) '\cap \mathcal{B}\left( H\right) ^{\mathcal U}\neq \mathbb C I$ flat?
                          The reverse implication is true.

                          Cite this as: AimPL: Set theory and C* algebras, available at http://aimpl.org/settheorycstar.