continuous-trace algebra

Let XX be a locally compact Hausdorff space. An algebra of continuous trace over XX is a C *C^*-algebra AA with dual space A^=X\hat{A} = X, such that Fell’s condition holds: for each x 0Xx_0\in X, there is aAa\in A such that x(a)x(a) is a rank-one projection for each xx in a neighbourhood of x 0x_0. This holds if and only if the set of all aAa\in A for which the map πtr(π(a)π(a) *)\pi\mapsto tr (\pi(a)\pi(a)^*) is finite and continuous on A^\hat{A} is dense in AA. A Fell algebra is a C *C^*-algebra AA which satisfies Fell’s condition but XX is not necessarily Hausdorff.

A continuous trace C *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 *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

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