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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.

##### MSC:
 46L80 $$K$$-theory and operator algebras (including cyclic theory) 46L55 Noncommutative dynamical systems 19K35 Kasparov theory ($$KK$$-theory)
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##### References:
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