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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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