nLab continuous-trace algebra

Description

Let $X$ be a locally compact Hausdorff space. An algebra of continuous trace over $X$ is a $C^*$-algebra $A$ with dual space $\hat{A} = X$, such that Fell’s condition holds: for each $x_0\in X$, there is $a\in A$ such that $x(a)$ is a rank-one projection for each $x$ in a neighbourhood of $x_0$. This holds if and only if the set of all $a\in A$ for which the map $\pi\mapsto tr (\pi(a)\pi(a)^*)$ is finite and continuous on $\hat{A}$ is dense in $A$. A Fell algebra is a $C^*$-algebra $A$ which satisfies Fell’s condition but $X$ is not necessarily Hausdorff.

A continuous trace $C^*$-algebra is, to some extent, an operator algebraic counterpart to the theory of Azumaya algebras. Dixmier-Douady class has been designed originally to give invariant of such operator algebras.

Literature

• Iain Raeburn, Dana Williams, Morita equivalence and continuous-trace $C^*$-algebras, AMS Monographs 60 (1998) xiv+327 pp.
• Jonathan Rosenberg, Continuous-trace algebras from the bundle theoretic point of view, J. Austral. Math. Soc. Ser. A 47 (1989), no. 3, 368–381 MR91d:46090 doi
• chapter 9 in Joachim Cuntz, Ralf Meyer, Jonathan M. Rosenberg, Topological and bivariant K-theory, Oberwolfach Seminars
• Alex Kumjian, Paul Muhly, Jean Renault, Dana Williams: The Brauer group of a locally compact groupoid, Amer. J. Math. 120 (1998) 901-954 ps

category: operator algebras, noncommutative geometry