Dixmier–Douady class - Imported Knowlpedia Content

Definition

Let $A$ be a separable continuous-trace $C^*$ -algebra with spectrum $X=\widehat A$ , and assume $X$ is paracompact. For a fixed separable infinite-dimensional Hilbert space $H$ , the stabilization of $A$ is locally a field of compact-operator algebras. Its transition automorphisms define a principal $PU(H)$ -bundle over $X$ , whose obstruction class

\delta(A)\in H^3(X;\mathbb Z)

is the Dixmier–Douady class of $A$ . Equivalently, it is the image in degree-three integral cohomology of the Čech cocycle obtained by lifting the projective-unitary transition functions locally to unitary operators. The class depends on the identification of the spectrum with $X$ Raeburn–Williams, Chapter 5.

Classification and vanishing

Over a fixed second-countable locally compact Hausdorff space $X$ , the assignment $A\mapsto\delta(A)$ classifies continuous-trace $C^*$ -algebras up to strong Morita equivalence: two such algebras are Morita equivalent over $X$ exactly when their Dixmier–Douady classes agree. Every class in $H^3(X;\mathbb Z)$ occurs Raeburn–Williams, Chapter 5.

The class vanishes exactly when $A$ is Morita equivalent over $X$ to $C_0(X)$ . In the separable stable setting, vanishing says that the compact-operator bundle is trivial, so the algebra is isomorphic to $C_0(X,\mathcal K)$ . Thus $\delta(A)$ measures twisting rather than the local fiber type, which is already fixed.

Cocycles and examples

Choose local trivializations with transition maps $g_{ij}:U_i\cap U_j\to PU(H)$ , and choose local unitary lifts $\widetilde g_{ij}$ . On triple overlaps,

\widetilde g_{ij}\widetilde g_{jk}\widetilde g_{ki} =c_{ijk}1

for circle-valued functions $c_{ijk}$ . Their Čech class in $H^2(X;\mathbb T)$ , transported by the exponential-sequence connecting map, is $\delta(A)\in H^3(X;\mathbb Z)$ . Changing lifts changes the cocycle by a coboundary.

The trivial field $C_0(X,\mathcal K)$ has class zero. More generally, a principal $PU(H)$ -bundle with nonzero class produces a section algebra that is locally indistinguishable from the trivial compact-operator field but is not Morita equivalent to $C_0(X)$ over $X$ . This global obstruction is the phenomenon isolated in the original Dixmier–Douady theory Dixmier–Douady, pp. 227–284.

Conventions and scope

Some formulations classify stable continuous-trace algebras up to $*$ -isomorphism, while others classify arbitrary continuous-trace algebras up to Morita equivalence. Stability, separability, and paracompactness determine which version applies. The unqualified class $\delta(A)$ should therefore be read together with its spectrum and the chosen classification setting.

References

Jacques Dixmier and Adrien Douady, “Champs continus d'espaces hilbertiens et de $C^*$ -algèbres,” Bulletin de la Société Mathématique de France 91 (1963), 227–284. Numdam DOI record and full text. Relevant: the obstruction and classification of locally trivial fields of elementary $C^*$ -algebras.
Iain Raeburn and Dana P. Williams, Morita Equivalence and Continuous-Trace $C^*$ -Algebras, Mathematical Surveys and Monographs 60, American Mathematical Society, 1998. AMS DOI record. Relevant: Chapter 5 on the Dixmier–Douady class, Morita classification, and stable classification.