<i>C</i><sup>∗</sup>-algebras associated with irrational rotations

Rieffel, Marc A.

doi:10.2140/pjm.1981.93.415

Cited by 446 publications

(230 citation statements)

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“…Algebraically, all this is in accord with Connes' interpretation of R/G ϑ and T 2 ϑ as a 'noncommutative space', i.e. a noncommutative C * -algebra A ϑ [C1,C2,C3,R1,Bl]; however, our ↑H n (T 2 ϑ ) has a trivial preorder, for n > 0.…”

Section: Introductionsupporting

confidence: 64%

Directed combinatorial homology and noncommutative tori (The breaking of symmetries in algebraic topology)

Grandis¹

2005

Math. Proc. Camb. Phil. Soc.

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Abstract. This is a brief study of the homology of cubical sets, with two main purposes.First, this combinatorial structure is viewed as representing directed spaces, breaking the intrinsic symmetries of topological spaces. Cubical sets have a directed homology, consisting of preordered abelian groups where the positive cone comes from the structural cubes.But cubical sets can also express topological facts missed by ordinary topology. This happens, for instance, in the study of group actions or foliations, where a topologically-trivial quotient (the orbit set or the set of leaves) can be enriched with a natural cubical structure whose directed cohomology agrees with Connes' analysis in noncommutative geometry. Thus, cubical sets can provide a sort of 'noncommutative topology', without the metric information of C * -algebras.MSC: 55U10, 55Nxx, 81R60.

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Section: Introductionsupporting

confidence: 64%

Directed combinatorial homology and noncommutative tori (The breaking of symmetries in algebraic topology)

Grandis¹

2005

Math. Proc. Camb. Phil. Soc.

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“…It effectively gives rise to the irrational rotation C * -algebras ( [15]). In fact these are quotient algebras of the C * -group algebra of G. Its completion G is the rational Heisenberg group.…”

Section: Characters Of the Discrete Heisenberg Group And Of Its Complmentioning

confidence: 99%

Characters of the discrete Heisenberg group and of its completion

Tandra¹,

Moran²

2004

Math. Proc. Camb. Phil. Soc.

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“…The Hall conductivity is just the pairing between this cyclic 2-cocycle and an idempotent in the algebra: the spectral projection of the Hamiltonian. A crucial rôle is played by the noncommutative torus [89].…”

Section: Introductionmentioning

confidence: 99%

An Introduction to Noncommutative Spaces and their Geometries

Landi¹

1997

Lecture Notes in Physics

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C^∗-algebras associated with irrational rotations

Cited by 446 publications

References 22 publications

Directed combinatorial homology and noncommutative tori (The breaking of symmetries in algebraic topology)

Directed combinatorial homology and noncommutative tori (The breaking of symmetries in algebraic topology)

Characters of the discrete Heisenberg group and of its completion

An Introduction to Noncommutative Spaces and their Geometries

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C∗-algebras associated with irrational rotations

Cited by 446 publications

References 22 publications

Directed combinatorial homology and noncommutative tori (The breaking of symmetries in algebraic topology)

Directed combinatorial homology and noncommutative tori (The breaking of symmetries in algebraic topology)

Characters of the discrete Heisenberg group and of its completion

An Introduction to Noncommutative Spaces and their Geometries

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C^∗-algebras associated with irrational rotations