The Thom isomorphism in the ``twice'' equivariant \(K\)-theory of \(C^*\)-bundles
Given a compact Lie group \(G\) acting on the compact space \(X\) and the \(C^*\)-algebra \(A\) continuously (by involutive automorphisms on \(A\)), let \(\text{Vect}_G(X; A)\) be the Banach category whose objects are \(GGA\)-bundles and whose morphisms are morphisms of \(\text{Vect}(X; A)\) commuting with the action of \(G\). A \(GGA\)-bundle is an \(A\)-bundle \(p: E\to X\) where \(E\) is also a \(G\)-space, so that the following conditions hold: (1) \(gp(e)= pg(e)\); (2) \(g: p^{-1}(x)\to p^{-1}(gx)\) is a \(GGA\)-linear mapping (i.e., \(g(e\cdot a)= g(e)\cdot g(a))\), \(\forall g\in G\), \(e\in E\) and \(x\in X\). Following the general scheme presented in \textit{M. Karoubi's} paper [Ann. Sci. Éc. Norm. Supér. (4) 1, No. 2, 161--270 (1968; Zbl 0194.24302)], the \(K\)-groups for \(X\) are defined: \[ \begin{aligned} \mathbb{K}^{p,q}_G(X, A) &= K^{p,q}(\text{Vect}_G(X, A)),\\ \mathbb{K}_G(X, A) &= K^{0,0}(\text{Vect}_G(X, A)).\end{aligned} \] Moreover, the induced Thom homomorphism \(\varphi_A\) in the corresponding \(K_G(:, A)\)-theory is defined. The main theorem gives a simple proof of the Thom isomorphism theorem: if \(X\) is a manifold, then \(\varphi_A\) is an isomorphism.
scientific article; zbMATH DE number 1964705