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 (Max (A) , 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): H ˇ j (Max (A) , Max (A / I)) ; Q) ≅ π 2 n - j - 1 (L c n (I)) Q for j < 2 n - 1 , where L c n (I) refers to the space of last columns. We then study the rational cohomological dimension cdim Q 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 Max (A) is metrizable and cdim Q Max (A) ≤ m , then (i) the rational homotopy group π k (G L n (A)) Q is stabilized if n ≥ ⌈ (m + k + 1) / 2 ⌉ and (ii) there exists a compact metrizable space X A with dim X A ≤ m such that A is embedded into the commutative C ∗ -algebra C (X A) such that π k (G L n (C (X A))) is rationally isomorphic to π k (G L n (A)) for each k ≥ 1 and π k (G L n (C (X A)) is stabilized for n ≥ ⌈ (m + k + 1) / 2 ⌉ . The main technical ingredient is a modified version of a classical theorem of Davie (Proc Lond Math Soc 23:31–52, 1971). [ABSTRACT FROM AUTHOR]