Rieffel deformation of homogeneous spaces

Pawel Lukasz Kasprzak

doi:10.1016/j.jfa.2010.08.009

Rieffel deformation of homogeneous spaces

Pawel Lukasz Kasprzak

Institut for Matematiske Fag

5 Citationer (Scopus)

Abstract

Let G₁⊂G be a closed subgroup of a locally compact group G and let X=G/G₁ be the quotient space of left cosets. Let X=(C₀(X),Δ_X) be the corresponding G-C^*-algebra where G=(C₀(G),Δ). Suppose that Γ is a closed abelian subgroup of G₁ and let ψ be a 2-cocycle on the dual group Γ̂. Let G^ψ be the Rieffel deformation of G. Using the results of the previous paper of the author we may construct G^ψ-C^*-algebra X^ψ - the Rieffel deformation of X. On the other hand we may perform the Rieffel deformation of the subgroup G1 obtaining the closed quantum subgroup G^1ψ⊂G^ψ, which in turn, by the results of S. Vaes, leads to the G^ψ-C^*-algebra G^ψ/G^1ψ. In this paper we show that G^ψ/G^1ψ≅X^ψ. We also consider the case where Γ⊂G is not a subgroup of G₁L, for which we cannot construct the subgroup G₁^ψ. Then generically X^ψ cannot be identified with a quantum quotient. What may be shown is that it is a G^ψ-simple object in the category of G^ψ-C^*-algebras.

Originalsprog	Engelsk
Tidsskrift	Journal of Functional Analysis
Vol/bind	260
Udgave nummer	1
Sider (fra-til)	146-153
ISSN	0022-1236
DOI	https://doi.org/10.1016/j.jfa.2010.08.009
Status	Udgivet - 1 jan. 2011

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10.1016/j.jfa.2010.08.009

Citationsformater

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title = "Rieffel deformation of homogeneous spaces",

abstract = "Let G1⊂G be a closed subgroup of a locally compact group G and let X=G/G1 be the quotient space of left cosets. Let X=(C0(X),ΔX) be the corresponding G-C*-algebra where G=(C0(G),Δ). Suppose that Γ is a closed abelian subgroup of G1 and let ψ be a 2-cocycle on the dual group {\^Γ}. Let Gψ be the Rieffel deformation of G. Using the results of the previous paper of the author we may construct Gψ-C*-algebra Xψ - the Rieffel deformation of X. On the other hand we may perform the Rieffel deformation of the subgroup G1 obtaining the closed quantum subgroup G1ψ⊂Gψ, which in turn, by the results of S. Vaes, leads to the Gψ-C*-algebra Gψ/G1ψ. In this paper we show that Gψ/G1ψ≅Xψ. We also consider the case where Γ⊂G is not a subgroup of G1L, for which we cannot construct the subgroup G1ψ. Then generically Xψ cannot be identified with a quantum quotient. What may be shown is that it is a Gψ-simple object in the category of Gψ-C*-algebras.",

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N2 - Let G1⊂G be a closed subgroup of a locally compact group G and let X=G/G1 be the quotient space of left cosets. Let X=(C0(X),ΔX) be the corresponding G-C*-algebra where G=(C0(G),Δ). Suppose that Γ is a closed abelian subgroup of G1 and let ψ be a 2-cocycle on the dual group Γ̂. Let Gψ be the Rieffel deformation of G. Using the results of the previous paper of the author we may construct Gψ-C*-algebra Xψ - the Rieffel deformation of X. On the other hand we may perform the Rieffel deformation of the subgroup G1 obtaining the closed quantum subgroup G1ψ⊂Gψ, which in turn, by the results of S. Vaes, leads to the Gψ-C*-algebra Gψ/G1ψ. In this paper we show that Gψ/G1ψ≅Xψ. We also consider the case where Γ⊂G is not a subgroup of G1L, for which we cannot construct the subgroup G1ψ. Then generically Xψ cannot be identified with a quantum quotient. What may be shown is that it is a Gψ-simple object in the category of Gψ-C*-algebras.

AB - Let G1⊂G be a closed subgroup of a locally compact group G and let X=G/G1 be the quotient space of left cosets. Let X=(C0(X),ΔX) be the corresponding G-C*-algebra where G=(C0(G),Δ). Suppose that Γ is a closed abelian subgroup of G1 and let ψ be a 2-cocycle on the dual group Γ̂. Let Gψ be the Rieffel deformation of G. Using the results of the previous paper of the author we may construct Gψ-C*-algebra Xψ - the Rieffel deformation of X. On the other hand we may perform the Rieffel deformation of the subgroup G1 obtaining the closed quantum subgroup G1ψ⊂Gψ, which in turn, by the results of S. Vaes, leads to the Gψ-C*-algebra Gψ/G1ψ. In this paper we show that Gψ/G1ψ≅Xψ. We also consider the case where Γ⊂G is not a subgroup of G1L, for which we cannot construct the subgroup G1ψ. Then generically Xψ cannot be identified with a quantum quotient. What may be shown is that it is a Gψ-simple object in the category of Gψ-C*-algebras.

U2 - 10.1016/j.jfa.2010.08.009

DO - 10.1016/j.jfa.2010.08.009

M3 - Journal article

SN - 0022-1236

VL - 260

SP - 146

EP - 153

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 1

ER -

Rieffel deformation of homogeneous spaces

Abstract

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