A DIXMIER-DOUADY THEORY FOR STRONGLY SELF-ABSORBING C∗-ALGEBRAS II: THE BRAUER GROUP

Abstract. We have previously shown that the isomorphism classes of orientable locally trivial fields of C∗-algebras over a compact metrizable space X with fiber D ⊗ K, where D is a strongly self-absorbing C∗-algebra, form an abelian group under the operation of tensor product. Moreover this group is isomorphic to the first group Ē1D(X) of the (reduced) generalized cohomology theory associated to the unit spectrum of topological K-theory with coefficients in D. Here we show that all the torsion elements of the group Ē1D(X) arise from locally trivial fields with fiber D ⊗Mn(C), n ≥ 1, for all known examples of strongly self-absorbing C∗-algebras D. Moreover the Brauer group generated by locally trivial fields with fiber D ⊗Mn(C), n ≥ 1 is isomorphic to Tor(Ē1D(X)). 1