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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 $\begin{split} K^i_\Gamma C(X) \cong KK^\Gamma_i(C_0(X), \mathbb C) \cong K_{i+1}(D^*_\Gamma(X)/C^*_\Gamma(X))\\ \to K_i(C^*_\Gamma(X)) \cong K_i(C^*_r(\Gamma))\cong KK_i(\mathbb C, C^*_r(\Gamma)).\end{split}$ .

##### MSC:
 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)
##### Keywords:
Paschke duality theory; analytic assembly map; C*-algebra
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