The Brauer group of graded continuous trace 𝐶*-algebras

Parker, Ellen Maycock

doi:10.1090/S0002-9947-1988-0946434-7

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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
MathSciNet review: 946434
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Abstract: Let 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 form a group, ${\operatorname{GBr} ^\infty }(X)$ , the infinite-dimensional graded Brauer group of . Techniques from algebraic topology are used to prove that ${\operatorname{GBr} ^\infty }(X)$ is isomorphic via an isomorphism 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 .

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