Approximability for times-invariant ultrafilters

Determine whether the Stone space associated with the statistical ideal \(\mathcal I_{\mathcal U}=\{A\subseteq\omega:d_{\mathcal U}(A)=0\}\) is approximable when the free ultrafilter \(\mathcal U\) is \(\times\)-invariant, and consequently determine whether \(c_{0,\mathcal I_{\mathcal U}}\) is complemented in \(\ell_\infty\) in this case.

Background

The paper proves that if a free ultrafilter $\mathcal U$ is not $\times$-invariant, then the Stone space $Stone(\mathcal P(\omega)/\mathcal I_{\mathcal U})$, equivalently the support of the associated density measure, is not approximable. By the cited characterization of complemented ideals, this implies that $c_{0,\mathcal I_{\mathcal U}}$ is not complemented in $\ell_\infty$.

For a $\times$-invariant ultrafilter, the measure on the quotient Boolean algebra is not countably additive, and the canonical embedding into the relevant metric ultraproduct is not onto. Thus, the identification used to establish non-approximability in the non-invariant case is unavailable, leaving the approximability and complementation behavior unresolved.