continuous-trace algebra

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.

- I. Raeburn, D. 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, Ralph 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 algebrasnoncommutative geometry

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