Dixmier-Douady Theory

Dixmier-Douady classification of continuous-trace $C^\ast$-algebras

Theorem(The Dixmier-Douady classification) . Let $A$ and $B$ be continuous-trace $C^\ast$-algebras with paracompact spectrum $X$. $A$ is Morita equivalent to $B$ over $X$ iff $\delta(A)=\delta(B)$ in $H^3(X;\mathbb{Z})$. Every element in $H^3(X;\mathbb{Z}$ is the Dixmier-Douady class of some continuous-trace algebra with spectrum $X$.

Bundles with fiber $\mathcal{K}(\mathcal{H})$

There is a theorem regarding the sheaf morphism given by $$ e:\underline{\mathbb{R}}\to \underline{\mathrm{U}(1)}: e_U(f) = \exp(2\pi i f), $$ where the sheaves are over $X$.

Theorem. The connecting homomorphisms $\Delta^n:H^n(X,\underline{\mathrm{U}(1)})\to H^{n+1}(X,\mathbb{Z})$ associated to the short exact sequence $$ 0 \to \underline{\mathbb{Z}} \to \underline{\mathbb{R}}\to \underline{\mathrm{U}(1)} $$ are isomorphisms for all $n>1$. For $n=0$, $\Delta$ induces an isomorphism of $C(X,U(1))/e(C(X,\mathbb{R}))$ onto $H^1(X,\mathbb{Z})$.

The Dixmier-Douady classification of bundles with fiber $\mathcal{K}(\mathcal{H})$ over a topological space $X$ is achieved by the following steps:

  1. Construct a map $\Delta : H^1(X,\underline{\mathrm{PU}(1)}) \to H^2(X,\underline{\mathrm{U}(1)})$.
  2. Apply the isomorphism $H^2(X,\underline{\mathrm{U}(1)})\cong H^3(X,\mathbb{Z})$ to get a class $\Delta([\alpha_{ij}])$ in the integral cohomology which classifies the original bundle up to isomorphism.