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Comparing analytic assembly maps. (English) Zbl 1014.46045
Denote by \(X\) a proper metric space. An operator \(T\) on the Hilbert space \(L^2(X)\) is called locally compact (resp., pseudo-local), denoted \(T\in C^*(X)\) (resp., \(T\in D^*(X)\)), iff for all \(f \in C_0(X)\), \(Tf\) and \(fT\) are (resp., \(fT-Tf\) is a) compact operator. The coarse assembly map is the homomorphism \(\mu : K_i(X) \cong K_{i+1}(D^*(X)/C^*(X)) \to K_i(C^*(X))\). In the case when there exists a \(\Gamma\)-invariant action of a discrete group \(\Gamma\) on \(X\), the analytic assembly map is \(\mu : KK^\Gamma(C_0(X), \mathbb C) \to KK^*(\mathbb C,C^*_r(\Gamma))\), which is a composite of two homomorphisms \[ KK_i^\Gamma(C_0(X),\mathbb C) \to KK_i(C_0(X) \rtimes_r \Gamma, C^*_r(\Gamma)) \to KK_i(\mathbb C,C^*_r(\Gamma)). \] The author presents another way to obtain the assembly map as \[ \begin{split} K^i_\Gamma C(X) \cong KK^\Gamma_i(C_0(X), \mathbb C) \cong K_{i+1}(D^*_\Gamma(X)/C^*_\Gamma(X))\\ \to K_i(C^*_\Gamma(X)) \cong K_i(C^*_r(\Gamma))\cong KK_i(\mathbb C, C^*_r(\Gamma)).\end{split} \] .

46L80 \(K\)-theory and operator algebras (including cyclic theory)
46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)
19K35 Kasparov theory (\(KK\)-theory)
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