The Brauer group of graded continuous trace 𝐶*-algebras

Parker, Ellen Maycock

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

The Brauer group of graded continuous trace $C^ *$-algebras
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Trans. Amer. Math. Soc. 308 (1988), 115-132 Request permission

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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