Covariant Fields of C<sup>*</sup>-Algebras under Rieffel Deformation

Belmonte, Fabián; Măntoiu, Marius

doi:10.3842/sigma.2012.091

Cited by 2 publications

(3 citation statements)

References 25 publications

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“…This allows us to conclude with Theorem 27 that C(M θ ) is isomorphic to the C * -algebra Γ(M/T 2 , B M θ ) of continuous sections of an upper semi-continuous C * -bundle B M θ → M/T 2 and that G(C(M θ ), H; J) acts by vertical automorphisms on B M θ . This also follows from the more general results of [2] showing that torus-covariant C(X)-algebras are deformed to torus-covariant C(X)-algebras. Here a torus-covariant algebra is a C(X)-algebra which carries an action of T 2 that commutes with C(X).…”

Section: Toric Noncommutative Manifolds a Less Trivial Example Is Gimentioning

confidence: 53%

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Localizing gauge theories from noncommutative geometry

Suijlekom

2016

Advances in Mathematics

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We recall the emergence of a generalized gauge theory from a noncommutative Riemannian spin manifold, viz. a real spectral triple (A, H, D; J). This includes a gauge group determined by the unitaries in the * -algebra A and gauge fields arising from a socalled perturbation semigroup which is associated to A. Our main new result is the interpretation of this generalized gauge theory in terms of an upper semi-continuous C * -bundle on a (Hausdorff) base space X. The gauge group acts by vertical automorphisms on this C * -bundle and can (under some mild conditions) be identified with the space of continuous sections of a group bundle on X. This then allows for a geometrical description of the group of inner automorphisms of A.We exemplify our construction by Yang-Mills theory and toric noncommutative manifolds and show that they actually give rise to continuous C * -bundles. Moreover, in these examples the corresponding inner automorphism groups can be realized as spaces of sections of group bundles that we explicitly determine. Contents arXiv:1411.6482v1 [math-ph] 24 Nov 2014 [a, [D, b]] = [Ja * J −1 , [D, b]] = 0, for a ∈ A J and b ∈ A.Example 10. In the case of a Riemannian spin manifold M with real structure J M given by charge conjugation, one checks that

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Section: Toric Noncommutative Manifolds a Less Trivial Example Is Gimentioning

confidence: 53%

“…If u = λI N is a multiple of the identity with λ ∈ U (1), this action is trivial. In fact, the group of automorphisms of A is the projective unitary group P U (N ) = U (N )/U (1), in concordance with (2).…”

Section: Spectral Triplesmentioning

confidence: 99%

Localizing gauge theories from noncommutative geometry

Suijlekom

2016

Advances in Mathematics

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“…To define the Rieffel deformation, one keeps the involution unchanged and introduces on A ∞ the product defined by oscillatory integrals (1) a…”

Section: Rieffel Deformationmentioning

confidence: 99%

Classification of irrational $Θ$-deformed CAR $C^*$-algebras

Kuźmin¹,

Turowska²

2020

Preprint

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Given a skew-symmetric matrix Θ we consider the universal enveloping C * -algebra CARΘ of the * -algebra generated by a1, . . . , an subject to relations a * i ai + aia * i = 1, a * i aj = e 2πiΘ i,j aja * i , aiaj = e −2πiΘ i,j ajai. We prove that CARΘ has a C(Kn)-structure, where Kn = [0, 1 2 ] n is the hypercube and describe the fibers. We classify irreducible representations of CARΘ in terms of irreducible representations of higherdimensional noncommutative tori. We prove that for a given irrational Θ1 there are only finitely many Θ2 such that CARΘ 1 ≃ CARΘ 2 . Namely, CARΘ 1 ≃ CARΘ 2 implies (Θ1)ij = ±(Θ2) σ(i,j) mod Z for a bijection σ of the set {(i, j) : i < j, i, j = 1, . . . , n}. For n = 2 it means that CAR θ 1 ≃ CAR θ 2 iff θ1 = ±θ2 mod Z.

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Covariant Fields of C^*-Algebras under Rieffel Deformation

Cited by 2 publications

References 25 publications

Localizing gauge theories from noncommutative geometry

Localizing gauge theories from noncommutative geometry

Classification of irrational $Θ$-deformed CAR $C^*$-algebras

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Covariant Fields of C*-Algebras under Rieffel Deformation

Cited by 2 publications

References 25 publications

Localizing gauge theories from noncommutative geometry

Localizing gauge theories from noncommutative geometry

Classification of irrational $Θ$-deformed CAR $C^*$-algebras

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Covariant Fields of C^*-Algebras under Rieffel Deformation