The Heisenberg group and \(K\)-theory.

*(English)*Zbl 0803.46076A unifying proof of Bott periodicity and that of the Connes isomorphism theorem is given using continuous fields of \(C^*\)-algebras. The last theorem asserts that for each \(C^*\)-dynamical system \((A,\alpha)\) there is a canonical isomorphism of Abelian groups \(K_ *(A\rtimes_ \alpha\mathbb{R})\to K_{*+1}(A)\). In the case of trivial action this reduces to the isomorphism \(K_ * SA\to K_{*+1} A\) of Bott periodicity (where \(SA\) is of course \(C_ 0(\mathbb{R})\otimes A\)). The main tools are:

1) Theorem 2.1 of Tomiyama and Lee, which gives a criterion, involving only the topology of the spectrum, for a \(C^*\)-algebra to arise from a continuous field of quotients. It appears to be the key to showing that many operations, when performed on a continuous field of \(C^*\)- algebras, give rise to another continuous field.

2) Theorem 3.1, which asserts that if the continuous field of \(C^*\)- algebras \(A_ h\), \(h\in [0,1]\) is trivial away from 0, then there exists a canonical homomorphism \(K_ * A_ 0\to K_ * A_ 1\), which is the identity in the case when the field is also trivial at 0. The most important example is the \(C^*\)-algebra of the Heisenberg group – it is the algebra of continuous sections vanishing at infinity of a continuous field of \(C^*\)-algebras \((A_ h)\), \(h\in \mathbb{R}\) and \(A_ 0\simeq C_ 0(\mathbb{R}^ 2)\), \(A_ h\simeq{\mathcal K}\) – the \(C^*\)-algebra of compact operators on \(L^ 2\mathbb{R}\), \(h\neq 0\). The key point is Theorem 4.5 (special case of Connes result), which states that the map \(K_ 0 A_ 0\to K_ 0 A_ 1\) of Theorem 3.1 in the case of Heisenberg group takes the Bott element of \(K_ 0 A_ 0= K_ 0(C_ 0(\mathbb{R}^ 2))\) into the positive generator of \(K_ 0 A_ 1= K_ 0{\mathcal K}= \mathbb{Z}\). The authors establish this property “…in as elementary a way as possible”.

(Two proofs are given. Remarkably, juxtaposing the proofs leads to a new proof of the Atiyah-Singer theorem!)

Now, proving Connes’ theorem, the authors include the natural epimorphism \(K_ 0 A\to K_ 1 A\rtimes_ \alpha\mathbb{R}\) into a graph of maps, three of which are associated with suspension and the remaining ones are of the type given by Theorem 3.1. Commutativity of the squares in the graph is established and the isomorphism desired follows, since it is included in the chain of maps with an identity as the composition.

1) Theorem 2.1 of Tomiyama and Lee, which gives a criterion, involving only the topology of the spectrum, for a \(C^*\)-algebra to arise from a continuous field of quotients. It appears to be the key to showing that many operations, when performed on a continuous field of \(C^*\)- algebras, give rise to another continuous field.

2) Theorem 3.1, which asserts that if the continuous field of \(C^*\)- algebras \(A_ h\), \(h\in [0,1]\) is trivial away from 0, then there exists a canonical homomorphism \(K_ * A_ 0\to K_ * A_ 1\), which is the identity in the case when the field is also trivial at 0. The most important example is the \(C^*\)-algebra of the Heisenberg group – it is the algebra of continuous sections vanishing at infinity of a continuous field of \(C^*\)-algebras \((A_ h)\), \(h\in \mathbb{R}\) and \(A_ 0\simeq C_ 0(\mathbb{R}^ 2)\), \(A_ h\simeq{\mathcal K}\) – the \(C^*\)-algebra of compact operators on \(L^ 2\mathbb{R}\), \(h\neq 0\). The key point is Theorem 4.5 (special case of Connes result), which states that the map \(K_ 0 A_ 0\to K_ 0 A_ 1\) of Theorem 3.1 in the case of Heisenberg group takes the Bott element of \(K_ 0 A_ 0= K_ 0(C_ 0(\mathbb{R}^ 2))\) into the positive generator of \(K_ 0 A_ 1= K_ 0{\mathcal K}= \mathbb{Z}\). The authors establish this property “…in as elementary a way as possible”.

(Two proofs are given. Remarkably, juxtaposing the proofs leads to a new proof of the Atiyah-Singer theorem!)

Now, proving Connes’ theorem, the authors include the natural epimorphism \(K_ 0 A\to K_ 1 A\rtimes_ \alpha\mathbb{R}\) into a graph of maps, three of which are associated with suspension and the remaining ones are of the type given by Theorem 3.1. Commutativity of the squares in the graph is established and the isomorphism desired follows, since it is included in the chain of maps with an identity as the composition.

Reviewer: G.A.Noskov (Omsk)

##### MSC:

46L80 | \(K\)-theory and operator algebras (including cyclic theory) |

46L55 | Noncommutative dynamical systems |

##### Keywords:

Bott periodicity; Connes isomorphism theorem; continuous fields of \(C^*\)-algebras; \(C^*\)-dynamical system; canonical isomorphism of Abelian groups; Atiyah-Singer theorem
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\textit{G. A. Elliott} et al., \(K\)-Theory 7, No. 5, 409--428 (1993; Zbl 0803.46076)

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##### References:

[1] | Atiyah, M. F.:K-Theory, Benjamin, New York, 1967. |

[2] | Atiyah, M. F.: Bott periodicity and the index of elliptic operators,Quart. J. Math. Oxford 19 (1968), 113-140. · Zbl 0159.53501 |

[3] | Blackadar, B.:K-Theory for Operator Algebras, Mathematical Sciences Research Institute Publications 5, Springer-Verlag, New York, 1986. · Zbl 0597.46072 |

[4] | Bott, R.: The stable homotopy of the classical groups,Ann. of Math. 70 (1959), 313-337. · Zbl 0129.15601 |

[5] | Connes, A.: Non-commutative differential geometry, Parts I and II,Inst. Hautes Etudes Sci. Publ. Math. 62 (1985), 257-360. · Zbl 0592.46056 |

[6] | Connes, A.: An analogue of the Thom isomorphism for crossed products of aC*-algebra by an action ofR,Adv. Math. 39 (1981), 31-55. · Zbl 0461.46043 |

[7] | Dixmier, J.: Sur les représentations des groupes de Lie nilpotents, IV,Canad. J. Math. 12 (1960), 324-352. · Zbl 0171.11702 |

[8] | Dixmier, J.:Les C*-algèbres et leurs représentations, Gauthier-Villars, Paris, 1964. · Zbl 0152.32902 |

[9] | Elliott, G. A.: Gaps in the spectrum of an almost periodic Schrödinger operator,C. R. Math. Rep. Acad. Sci. Canada 4 (1982), 255-259. · Zbl 0516.46048 |

[10] | Elliott, G. A. Natsume, T., and Nest, R.: Cyclic cohomology of crossed products withR,Acta Math. 160 (1988), 285-305. · Zbl 0655.46054 |

[11] | Elliott, G. A., Natsume, T., and Nest, R.: The Atiyah-Singer index theorem as passage to the classical limit in quantum mechanics, in preparation. · Zbl 0876.58047 |

[12] | Elliott, G. A., and Olesen, D.: A simple proof of the Dauns-Hofmann theorem,Math. Scand. 34 (1974), 231-234. · Zbl 0289.46042 |

[13] | Fack, T. and Skandalis, G.: Connes’ analogue of the Thom isomorphism for the Kasparov groups,Invent. Math. 64 (1981), 7-14. · Zbl 0482.46043 |

[14] | Howe, R.: On representations of discrete, finitely generated, torsion-free, nilpotent groups,Pacific J. Math. 73 (1977), 281-305. · Zbl 0387.22005 |

[15] | Lee, R.-Y.: On theC*-algebras of operator fields,Indiana Univ. Math. J. 25 (1976), 303-314. · Zbl 0322.46062 |

[16] | Pedersen, G. K.:C*-Algebras and their Automorphism Groups, Academic Press, London, 1979. · Zbl 0416.46043 |

[17] | Rieffel, M. A.: Continuous field ofC*-algebras coming from group cocycles and actions,Math. Ann. 283 (1989), 631-643. · Zbl 0646.46063 |

[18] | Rosenberg, J.: Homological invariants of extensions ofC*-algebras, in R. V. Kadison (ed),Operator Algebras and Applications, Proceedings of Symposia in Pure Mathematics 38, American Mathematical Society, Providence, 1982, Part 1, pages 35-75. |

[19] | Takai, H.: On a duality for crossed products ofC*-algebras,J. Funct. Anal. 19 (1975), 25-39. · Zbl 0295.46088 |

[20] | Tomiyama, J.: Topological representation ofC*-algebras,Tôhoku Math. J. 14 (1962), 187-204. · Zbl 0216.16201 |

[21] | Voiculescu, D.: Remarks on the singular extension in theC*-algebra of the Heisenberg group,J. Operator Theory 5 (1981), 147-170. · Zbl 0476.22008 |

[22] | Zeller-Meier, G.: Produits croisés d’uneC*-algèbre par un groupe d’automorphismes,J. Math. Pures Appl. 47 (1968), 101-239. · Zbl 0165.48403 |

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