Cohomological stabilization of maximal ideal spaces in Banach algebras (Q1340007)

The author proves that for a complex commutative Banach algebra \(A\) with finite dimensional character space \(X\), the Bass stable rank condition \(\text{sr } A\leq n\) is equivalent to a certain stabilization property for Čech cohomology sequences of pairs \((X,Z)\), where \(Z\) is the hull of an arbitrary closed ideal. In particular, the \(k\)-cohomology groups of \(X\), \(Z\) and \((X,Z)\) vanish when \(k\geq 2n+1\). Using this result and Michael's continuous selection theory he improves previous estimations for \(\text{sr } {\mathcal C} (Y,A)\), where \({\mathcal C} (Y,A)\) denotes the algebra of continuous functions from a finite dimensional compact space \(Y\) into \(A\).