Rational stabilization and maximal ideal spaces of commutative Banach algebras

For a unital commutative Banach algebra A and its closed ideal I, we study the relative Čech cohomology of the pair \((\mathrm {Max}(A),\mathrm {Max}(A/I))\) of maximal ideal spaces and show a relative version of the main theorem of Lupton et al. (Trans Amer Math Soc 361:267–296, 2009): \(\check{\mathrm {H}}^{j}(\mathrm {Max}(A),\mathrm {Max}(A/I));{\mathbb {Q}}) \cong \pi _{2n-j-1}(Lc_{n}(I))_{{\mathbb {Q}}}\) for \(j < 2n-1\), where \(Lc_{n}(I)\) refers to the space of last columns. We then study the rational cohomological dimension \(\mathrm {cdim}_{\mathbb Q}\mathrm {Max}(A)\) for a unital commutative Banach algebra and prove an embedding theorem: if A is a unital commutative semi-simple regular Banach algebra such that \(\mathrm {Max}(A)\) is metrizable and \(\mathrm {cdim}_{{\mathbb {Q}}}\mathrm {Max}(A) \le m\), then (i) the rational homotopy group \(\pi _{k}(GL_{n}(A))_{{\mathbb {Q}}}\) is stabilized if \(n \ge \lceil (m+k+1)/2\rceil \) and (ii) there exists a compact metrizable space \(X_A\) with \(\dim X_{A} \le m\) such that A is embedded into the commutative \(C^*\)-algebra \(C(X_{A})\) such that \(\pi _{k}(GL_{n}(C(X_{A})))\) is rationally isomorphic to \(\pi _{k}(GL_{n}(A))\) for each \(k\ge 1\) and \(\pi _{k}(GL_{n}(C(X_{A}))\) is stabilized for \(n \ge \lceil (m+k+1)/2 \rceil \). The main technical ingredient is a modified version of a classical theorem of Davie (Proc Lond Math Soc 23:31–52, 1971).