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The Brauer group of graded continuous trace $ C\sp *$-algebras


Author: Ellen Maycock Parker
Journal: Trans. Amer. Math. Soc. 308 (1988), 115-132
MSC: Primary 46L05; Secondary 16A16, 22D25, 55R10
DOI: https://doi.org/10.1090/S0002-9947-1988-0946434-7
MathSciNet review: 946434
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Abstract: Let $ X$ be a locally compact Hausdorff space. The graded Morita equivalence classes of separable, $ {{\mathbf{Z}}_2}$-graded, continuous trace $ {C^{\ast}}$-algebras which have spectrum $ X$ form a group, $ {\operatorname{GBr} ^\infty }(X)$, the infinite-dimensional graded Brauer group of $ X$. Techniques from algebraic topology are used to prove that $ {\operatorname{GBr} ^\infty }(X)$ is isomorphic via an isomorphism $ w$ to the direct sum $ \check{H}^1(X; \underline{\mathbf{Z}}_2) \oplus \check{H}^3 (X; \underline{\mathbf{Z}})$. The group $ {\operatorname{GBr} ^\infty }(X)$ includes as a subgroup the ungraded continuous trace $ {C^{\ast}}$-algebras, and the Dixmier-Douady invariant of such an ungraded $ {C^{\ast}}$-algebra is its image in $ \check{H}^3 (X; \underline{\mathbf{Z}})$ under $ w$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1988-0946434-7
Keywords: $ {C^{\ast}}$-algebra, continuous trace, graded, Brauer group, Dixmier-Douady invariant, fiber bundle, Morita equivalence
Article copyright: © Copyright 1988 American Mathematical Society

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