Toeplitz operators on noncommutative spheres and an index theorem

Natsume, Toshikazu; Olsen, Catherine L.

doi:10.1512/iumj.1997.46.1152

Cited by 12 publications

(45 citation statements)

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“…In this sense, one may think of a noncommutative C * -algebra with unit as a dual object to some nebulous "compact noncommutative space." This type of reasoning is more grounded when a noncommutative C * -algebra A is obtained from some C(X) by a deformation procedure, such as the q-deformed ( [26]) and θ-deformed ( [16], [5], [21]) noncommutative spheres. In both of these cases, a C * -algebraic analogue of the Borsuk-Ulam theorem holds, where oddness of functions is replaced with discussion of a natural Z/2Z action on the algebra, which we call the antipodal action.…”

Section: Introductionmentioning

confidence: 99%

Free actions on C*-algebra suspensions and joins by finite cyclic groups

Passer¹

2018

Indiana Univ. Math. J.

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We present a proof for certain cases of the noncommutative Borsuk-Ulam conjectures proposed by Baum, D ' abrowski, and Hajac. When a unital C * -algebra A admits a free action of Z/kZ, k ≥ 2, there is no equivariant map from A to the C * -algebraic join of A and the compact "quantum" group C(Z/kZ). This also resolves D ' abrowski's conjecture on unreduced suspensions of C * -algebras. Finally, we formulate a different type of noncommutative join than the previous authors, which leads to additional open problems for finite cyclic group actions.

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Section: Introductionmentioning

confidence: 99%

Free actions on C*-algebra suspensions and joins by finite cyclic groups

Passer¹

2018

Indiana Univ. Math. J.

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“…When a Rieffel deformation procedure ( [11]) is applied to the function algebra C(S k ) of a sphere S k , the resulting C * -algebra admits a succinct presentation ( [7], [4]). Definition 1.1.…”

Section: Introductionmentioning

confidence: 99%

Anticommutation in the presentations of theta-deformed spheres

Passer¹

2017

Journal of Mathematical Analysis and Applications

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“…The conditions were determined through the computation of various examples, as in [17,16], chief among them odd-dimensional θ-deformed spheres (defined in [13,15]) and twisted versions thereof. In section 2 we consider more restrictive assumptions that guarantee nonexistence of equivariant maps from A to J(A, β).…”

Section: There Does Not Exist Amentioning

confidence: 99%

“…Finally, following the K-theoretic computations in [18,16], we see that θ-deformed spheres, and certain twisted unreduced suspensions thereof, admit twisted Borsuk-Ulam theorems. First, we recall the definitions, as in [13,15,16]. In both cases, the antipodal action on the (2n−3)-dimensional ω-sphere induces the antipodal action on the (twisted) join, in the usual way.…”

Section: Nonexistencementioning

confidence: 99%

Triviality of equivariant maps in crossed products and matrix algebras

Passer

2018

The Quarterly Journal of Mathematics

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We consider a "twisted" noncommutative join procedure for unital C * -algebras which admit actions by a compact abelian group G and its discrete abelian dual Γ, so that we may investigate an analogue of Baum-D ' abrowski-Hajac noncommutative Borsuk-Ulam theory in the twisted setting. Namely, under what conditions is it guaranteed that an equivariant map φ from a unital C * -algebra A to the twisted join of A and C * (Γ) cannot exist? This pursuit is motivated by the twisted analogues of even spheres, which admit the same K0 groups as even spheres and have an analogous Borsuk-Ulam theorem that is detected by K0, despite the fact that the objects are not themselves deformations of a sphere. We find multiple sufficient conditions for twisted Borsuk-Ulam theorems to hold, one of which is the addition of another equivariance condition on φ that corresponds to the choice of twist. However, we also find multiple examples of equivariant maps φ that exist even under fairly restrictive assumptions. Finally, we consider an extension of unital contractibility (in the sense of D ' abrowski-Hajac-Neshveyev) "modulo k."

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Toeplitz operators on noncommutative spheres and an index theorem

Cited by 12 publications

References 0 publications

Free actions on C*-algebra suspensions and joins by finite cyclic groups

Free actions on C*-algebra suspensions and joins by finite cyclic groups

Anticommutation in the presentations of theta-deformed spheres

Triviality of equivariant maps in crossed products and matrix algebras

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