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Banach algebras and rational homotopy theory. (English) Zbl 1163.55006
Let \(A\) be a unital commutative Banach algebra. Denote by Max \((A)\) the set of maximal ideals of \(A\) topologized with the relative weak\(^*\)-topology and by \(GL_n(A)\) the group of invertible \(n\times n\) matrices with coefficients in \(A\). Denote finally by \(Lc_n(A)\) the orbit of the vector \(e_n\) under the natural action of \(GL_n(A)\). The main result of the paper is the complete determination of the rational homotopy type of \(GL_n(A)\). More precisely, denote \(\tilde V_{h,j} = \tilde H^{2j-1-h} (\text{Max} (A);\mathbb Q)\). Then there is a natural rational H-equivalence \[ GL_n(A) \simeq_{\mathbb Q} \prod_{j=1}^n\prod_{h=1}^{2j-1} K(\tilde V_{h,j},h)\,. \] The rational cohomology groups of Max \((A)\) are determined by the homotopy groups of \(Lc_n(A)\) via the isomorphisms
\[ \tilde H^s(\text{Max} (A);\mathbb Q) \cong \pi_{2n-1-s} (Lc_n(A))\otimes\mathbb Q\,, \text{for}\,\, n>\frac{1}{2}s+1\,. \]
The results are based on the Theorem of Davies that gives an homotopy equivalence between \(GL_n(A)\) and the space of continuous functions \(F(\text{Max}(A), GL_n(\mathbb C))\).

MSC:
55P62 Rational homotopy theory
46L85 Noncommutative topology
46J05 General theory of commutative topological algebras
54C35 Function spaces in general topology
55P15 Classification of homotopy type
55P45 \(H\)-spaces and duals
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