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The Homotopy 3-Type of Abelian C*-Algebras

Gregory Faurot, Giovanni Ferrer

Abstract

We compute the homotopy groups at each unital abelian C*-algebra $C(T)$ in the Morita $3$-category of abelian C*-algebras, C*-algebras with central maps, C*-correspondences, and adjointable bimodule maps. We describe these groups in terms of the topological data of the underlying compact Hausdorff space $T$. We also compute the actions of the first homotopy group on the second and third homotopy groups in terms of these topological invariants of $T$.

The Homotopy 3-Type of Abelian C*-Algebras

Abstract

We compute the homotopy groups at each unital abelian C*-algebra in the Morita -category of abelian C*-algebras, C*-algebras with central maps, C*-correspondences, and adjointable bimodule maps. We describe these groups in terms of the topological data of the underlying compact Hausdorff space . We also compute the actions of the first homotopy group on the second and third homotopy groups in terms of these topological invariants of .

Paper Structure

This paper contains 13 sections, 46 theorems, 99 equations.

Table of Contents

  1. Background
  2. Imprimitivity Bimodules
  3. Spectra & Dauns--Hofmann
  4. Continuous-Trace C*-Algebras
  5. C*-categories
  6. Duality for Abelian C*-Algebras
  7. Computing the homotopy groups of AbC*Alg
  8. Results for the zeroth and the first homotopy groups
  9. Results for $\pi_2$ and $\pi_3$
  10. Actions on 2nd and 3rd homotopy groups
  11. Actions by Cohomology
  12. Actions by Homeomorphisms
  13. Serre--Swan duality

Key Result

Theorem A

Let $C(T)$ be an abelian C*-algebra. Then the homotopy groups at $C(T)$ in $\mathsf{AbC^*Alg}$ are as follows: Furthermore, if $T$ has the homotopy type of a CW-complex,

Theorems & Definitions (105)

  • Theorem A
  • Theorem B
  • Definition A
  • Remark B
  • Definition C
  • Definition D
  • Definition E
  • Definition F
  • Remark G
  • Definition H
  • ...and 95 more