#### Document Type

Article

#### Abstract

Let *A* be a unital commutative Banach algebra with maximal ideal space Max(*A*). We determine the rational H-type of GL_{n}(*A*), the group of invertible *n* x *n* matrices with coefficients in *A*, in terms of the rational cohomology of Max(*A*). We also address an old problem of J. L. Taylor. Let Lc_{n}(*A*) denote the space of "last columns" of GL_{n}(*A*). We construct a natural isomorphism

*Ȟ*^{s}(Max(*A*);ℚ) ≅ π_{2n-1-s}(Lc_{n}(*A*)) ⊗ ℚ

for *n* > ½*s*+1 which shows that the rational cohomology groups of Max(*A*) are determined by a topological invariant associated to *A*. As part of our analysis, we determine the rational H-type of certain guage groups *F*(*X,G*) for *G* a Lie group or, more generally, a rational H-space.

#### Disciplines

Algebra | Analysis | Geometry and Topology

#### Recommended Citation

G. Lupton, N. C. Phillips, C. L. Schochet and S. B. Smith, Banach algebras and rational homotopy theory, *Transactions of the American Mathematical Society* **361.1** (2009), 267-295.

#### Included in

Algebra Commons, Analysis Commons, Geometry and Topology Commons

## Comments

First published in

Transactions of the American Mathematical Societyin 361.1 (January 2009, http://www.ams.org/journals/tran/2009-361-01/S0002-9947-08-04477-2/), published by the American Mathematical Society.The research of the second author was partially supported by NSF grant DMS 0302401.

The authors would like to thank Daniel Isaksen and Jim Stasheff for helpful discussions.