Continuous-trace C*-algebra - Imported Knowlpedia Content

Definition

Let $A$ be a $C^*$ -algebra with Hausdorff spectrum $\widehat A$ of unitary-equivalence classes of irreducible representations. For $a\in A_+$ , define

\widehat a([\pi])=\operatorname{Tr}(\pi(a)).

The element $a$ has continuous trace when this function is finite and continuous on $\widehat A$ . The algebra $A$ is a continuous-trace $C^*$ -algebra when its continuous-trace positive elements are dense in $A_+$ . Continuity uses the canonical operator trace in each irreducible representation, while density makes such elements detect the whole algebra. This is a property of $A$ , not one representation.

Local compact-operator structure

Continuous-trace algebras are type I and, locally over their spectrum, are strongly Morita equivalent to commutative algebras $C_0(U)$ . Under the usual separability and paracompactness hypotheses, their stabilizations are section algebras of locally trivial bundles of compact-operator algebras. This is the geometric form developed in Raeburn–Williams, Chapters 4–5.

Consequently a continuous-trace algebra is a particularly regular continuous field of $C^*$ -algebras. Its fibers in irreducible representations are compact operators rather than arbitrary type I algebras.

Morita invariance

The continuous-trace property is preserved by strong Morita equivalence. The spectrum is carried along by the Rieffel correspondence, while the local compact-operator models are unchanged up to stabilization Raeburn–Williams, Chapters 3–5. This makes continuous-trace algebras natural noncommutative analogues of spaces equipped with a twisting class.

Examples and non-examples

For a locally compact Hausdorff space $X$ , both $C_0(X)$ and $C_0(X,\mathcal K(H))$ have continuous trace. The latter has spectrum $X$ , and positive finite-rank sections supply a dense family of continuous-trace elements.

If $H$ is infinite-dimensional, $B(H)$ does not have continuous trace. Its positive trace-class operators are not norm-dense in $B(H)_+$ ; equivalently, its identity has infinite trace and cannot be approximated in norm by compact finite-trace elements.

References

I. Raeburn and D. P. Williams, Morita Equivalence and Continuous-Trace $C^*$ -Algebras, Mathematical Surveys and Monographs 60, American Mathematical Society, 1998. DOI record. Relevant: Chapters 4–5 on continuous trace, local compact-operator models, and Morita invariance.
J. Dixmier, $C^*$ -Algebras, North-Holland Mathematical Library 15, North-Holland, 1977. Publisher record. Relevant: Chapter 10 on continuous fields and continuous-trace algebras.