An Equivariant Brauer Group and Actions of Groups onC*-Algebras

Crocker, D.; Kumjian, Alex; Raeburn, Iain; Williams, Dana P.

doi:10.1006/jfan.1996.3010

Cited by 48 publications

(90 citation statements)

References 31 publications

(58 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…see [5]). On the other hand, given ω ∈ Z 2 (G, T), then we can construct full and reduced twisted group algebras C * (G, ω) and C * r (G, ω) as follows: The twisted convolution algebra ℓ 1 (G, ω) is defined as the vector space of all summable complex functions on G with convolution and involution given by…”

Section: Some Applicationsmentioning

confidence: 96%

“…The main contribution of this paper is to extend this result by stating and proving a twisted version of it, namely that if C 0 (X, δ) is a continuous trace algebra, i.e., the section algebra of a locally trivial G-equivariant bundle of elementary C*-algebras representing an element δ of the Brauer group Br G (X) (see [5]), then C 0 (X, δ −1 ) ⋊ G is Poincaré dual to C 0 (X, δ)⊗ C 0 (X) C τ (X) ⋊ G, where δ −1 is the inverse of δ in the Brauer group. To prove this, we need to go back to the argument of the previous paragraph.…”

mentioning

confidence: 98%

See 1 more Smart Citation

KK-theoretic duality for proper twisted actions

2007

View full text Add to dashboard Cite

Abstract. Let A be a smooth continuous trace algebra, with a Riemannian manifold spectrum X, equipped with a smooth action by a discrete group G such that G acts on X properly and isometrically. Then A −1 ⋊G is KK-theoretically Poincaré dual to`A⊗ C 0 (X) Cτ (X)´⋊G, where A −1 is the inverse of A in the Brauer group of Morita equivalence classes of continuous trace algebras equipped with a group action. We deduce this from a strengthening of Kasparov's duality theorem. As applications we obtain a version of the above Poincaré duality with X replaced by a compact G-manifold M and Poincaré dualities for twisted group algebras if the group satisfies some additional properties related to the Dirac dual-Dirac method for the Baum-Connes conjecture.

show abstract

Section: Some Applicationsmentioning

confidence: 96%

mentioning

confidence: 98%

KK-theoretic duality for proper twisted actions

2007

View full text Add to dashboard Cite

show abstract

“…Alternatively, it can be read off from Theorem 4.3.3, since any two classes in ker M mapping to the same δ ∈ H 3 (X, Z) differ by the image under ξ of something in ker a. Thus they differ by the image under ξ of an Z-valued cocycle, which is trivial since such a cocycle exponentiates to the trivial cocycle with values in T, and this is all that is used in the construction of ξ in [12]. Finally, if [CT (X, δ), α] has trivial Mackey obstruction, then as explained in [26, §1], CT (X, δ) ⋊ α G has continuous trace and has spectrum which is another principal torus bundle over Z (for the dual torus, G divided by the dual lattice).…”

Section: 4mentioning

confidence: 99%

T-Duality for Torus Bundles with H-Fluxes via Noncommutative Topology

Mathai

Rosenberg

2004

Commun. Math. Phys.

140

326

View full text Add to dashboard Cite

Abstract. It is known that the T-dual of a circle bundle with H-flux (given by a Neveu-Schwarz 3-form) is the T-dual circle bundle with dual H-flux. However, it is also known that torus bundles with H-flux do not necessarily have a T-dual which is a torus bundle. A big puzzle has been to explain these mysterious "missing T-duals." Here we show that this problem is resolved using noncommutative topology. It turns out that every principal T 2 -bundle with H-flux does indeed have a T-dual, but in the missing cases (which we characterize), the T-dual is non-classical and is a bundle of noncommutative tori. The duality comes with an isomorphism of twisted K-theories, just as in the classical case. The isomorphism of twisted cohomology which one gets in the classical case is replaced by an isomorphism of twisted cyclic homology.

show abstract

“…We abbreviate notation for the K(L 2 (N))-valued kernels by setting 10) so that the equivariance condition can be given explicitly as 11) or, replacing the first two arguments, as…”

Section: The Twisted Crossed Product Algebramentioning

confidence: 99%

Nonassociative Tori and Applications to T-Duality

Bouwknegt

Hannabuss²,

Mathai

2006

Commun. Math. Phys.

118

229

View full text Add to dashboard Cite

Abstract. In this paper, we initiate the study of C * -algebras A endowed with a twisted action of a locally compact Abelian Lie group G, and we construct a twisted crossed product A⋊G, which is in general a nonassociative, noncommutative, algebra. The duality properties of this twisted crossed product algebra are studied in detail, and are applied to T-duality in Type II string theory to obtain the T-dual of a general principal torus bundle with general H-flux, which we will argue to be a bundle of noncommutative, nonassociative tori. We also show that this construction of the T-dual includes all of the special cases that were previously analysed.

show abstract

An Equivariant Brauer Group and Actions of Groups onC*-Algebras

Cited by 48 publications

References 31 publications

KK-theoretic duality for proper twisted actions

KK-theoretic duality for proper twisted actions

T-Duality for Torus Bundles with H-Fluxes via Noncommutative Topology

Nonassociative Tori and Applications to T-Duality

Contact Info

Product

Resources

About