Rational stabilization and maximal ideal spaces of commutative Banach algebras

Abstract

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).