Forcing Absoluteness of Minimal and Maximal C∗-Tensor Products

We answer an AIM question about whether the minimal or maximal \(C^*\) -tensor product commutes with forcing. We use the standard convention that a ground-model \(C^*\) -algebra is replaced in the extension by the metric completion of its old normed \(*\) -algebra. Under this convention, both \(\otimes_{\min}\) and \(\otimes_{\max}\) commute canonically with every set-forcing extension, for arbitrary ground-model factors; no separability, density-character, or cardinal-preservation hypothesis is needed. The minimal case follows by extension of faithful spatial representations. For the maximal case, any putative larger new norm would give finitely satisfiable old semialgebraic \(C^*\) -seminorm constraints. Quantifier elimination for real closed fields and compactness inside the ground model then produce an old \(C^*\) -seminorm exceeding the old universal norm, a contradiction.