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A Bott periodicity theorem for infinite dimensional Euclidean space. (English) Zbl 0911.46040
For a compact group \(G\) and a compact space \(X\), the equivariant K-theory \(K^0_G\) is the Grothendieck group of complex \(G\)-vector bundles on \(X\). The \(K^1_G(X) = K^0(SX)\) is the \(K^0_G\) over the suspension \(SX\) of \(X\). The corresponding reduced equivariant \(K\)-theory \(K^*_G(X):= K^0_G(X) \oplus K^1_G(X)\) for locally compact spaces is defined as the kernel of the natural projection map \(K^*_G(X) := \ker\{K^*_G(\tilde{X}) \to K^*_G(pt) \}\), where \(\tilde{X}\) denotes the one-point compactification of \(X\) and \(pt\) - the one-point set. The main result in the classical equivariant theory [G. Segal, Inst. Hautes Étud. Sci.,Publ. Math. 34, 129-151 (1968; Zbl 0199.26202)] is the so called Bott periodicity theorem asserting that for a finite dimensional Euclidean space \(E\), the equivariant \(K\)-theory of its tangent space \(TE\) is isomorphic to the same ones of the one-point space, \(K^*_G(TE) \cong K^*_G(pt)\). This isomorphism can be emphasized in the context of crossed products of \(C^*\)-algebras and groups as the isomorphism between equivariant \(K^*(C_0(TE)\rtimes G) \cong K^*_G(TE) \cong K^*_G(pt)\).
In the paper under review, the authors prove this kind theorem for infinite-dimensional Euclidean spaces \(E\). Their main theorem asserts that if \(G\) is a countable discrete group and \(E\) is a countably infinite dimensional Euclidean space then there is also an isomorphism of Abelian groups \(K^G_*(SC(E)):= K^*(SC(E)\rtimes G) \cong K^G_*(SC(0)):= K^*(SC(0)\rtimes G),\) where \(C(E)\) is the \(\mathbb{Z}/(2)\)-graded \(C^*\)-algebra \(C_0(E,\text{ Cliff}(E))\) of continuous, \(\text{ Cliff}(E)\)-valued functions on \(E\) vanishing at infinity, with \(\mathbb{Z}/(2)\)-grading induced from the Clifford algebra \(\text{ Cliff}(E)\) and \(SC(E)\) is defined as the direct limit \(\varinjlim_{E_\alpha \subset E} SC(E_\alpha)\) over the directed system of all infinite dimensional Euclidean subspaces \(E_\alpha\) of \(E\), with \(SC(E_\alpha) := C_0(\mathbb{R}) \widehat{\otimes} C(E_\alpha)\). This isomorphism has many applications to index theory and the famous Novikov conjecture on homotopy invariance of higher signatures in topology.

46L80 \(K\)-theory and operator algebras (including cyclic theory)
46L55 Noncommutative dynamical systems
19K35 Kasparov theory (\(KK\)-theory)
Full Text: DOI
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