The Heisenberg group and \(K\)-theory
type: Item
instance of: scholarly article
title: The Heisenberg group and \(K\)-theory
zbMATH Open document ID: 0803.46076
DOI: 10.1007/BF00961535
author: Toshikazu Natsume
author: Ryszard Nest
author: George A. Elliott
published in: \(K\)-Theory
publication date: 1995-01-02T00:00:00Z
review text: A 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 ``\dots 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 auth…
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 46L55
zbMATH DE Number: 529890
zbMATH Keywords: Bott periodicity
zbMATH Keywords: Connes isomorphism theorem
zbMATH Keywords: continuous fields of \(C^*\)-algebras
zbMATH Keywords: \(C^*\)-dynamical system
zbMATH Keywords: canonical isomorphism of Abelian groups
zbMATH Keywords: Atiyah-Singer theorem
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00961535
OpenAlex ID: W2045522200
description: scientific article; zbMATH DE number 529890