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The Heisenberg group and $$K$$-theory. (English) Zbl 0803.46076
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 “…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
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