THE BRAUER GROUP OF GRADED CONTINUOUS TRACE C*-ALGEBRAS

ELLEN MAYCOCK PARKER, Purdue University

Abstract

In this thesis, separable, (,2)-graded, continuous trace C*-algebras which are homogeneous of infinite degree are classified. The graded isomorphism classes of such algebras, whose spectra are all the same locally compact Hausdorff space X, form a group called the infinite-dimensional graded Brauer group of X, GBr('(INFIN))(X). Techniques from algebraic topology are used to prove that GBr('(INFIN))(X) is isomorphic via an isomorphism w to the direct sum (')H('1) (X; (,2)) (CRPLUS) (')H('3) (X; ), where (')H*(X; G) denotes the Cech cohomology of X with coefficients in the sheaf of germs of continuous functions from X to G when G is a topological group. The group GBr('(INFIN))(X) includes as a subgroup the ungraded continuous trace C*-algebras, and the Dixmier-Douady invariant of such an ungraded C*-algebra is its image in (')H('3)(X; ) under w. There is a canonical way to construct a graded fiber bundle over X with fiber H(,gr)(H), the graded compact operators on an infinite-dimensional Hilbert space H, from an element of GBr('(INFIN))(X). The correspondence between graded C*-algebras and graded fiber bundles allows the finite-dimensional cases to be included in GBr('(INFIN))(X). It is shown that the image in (')H('1)(X; (,2)) of an algebra A in GBr('(INFIN))(X) under w is the obstruction to the grading automorphism of A being an inner automorphism. In addition, a definition of graded Morita equivalence is given as an alternate equivalence relation to determine GBr('(INFIN))(X). Finally, the thesis contains applications to duality and K-theory. A K-theory twisted by elements of GBr('(INFIN))(X) is defined. The concept of a Spanier-Whitehead type dual to a graded continuous trace C*-algebra is explored. It is in the context of KK-theory that this classification of graded continuous trace C*-algebras is especially useful.

Degree

Ph.D.

Subject Area

Mathematics

COinS