\(C^*\)-algebra homomorphisms and \(KK\)-theory
For an algebra \(A\) of the so-called class AH, (i.e. which can be presented as the inductive limit of an inductive system \[ P_1M_n({\mathbb C}(X_1) \to P_2M_n({\mathbb C}(X_2) \to \dots , \] where \(X_i\) are finite CW-complexes such that \(\sup\dim X_i < \infty\), \(P_i \in M_{n_i}({\mathbb C}(X_i))\) are projections and for a finite CW-complex \(X\), which is a disjoint union \(X_1 \cup X_2 \cup \dots \cup X_n\) of connected CW-complexes,) the author proves (Theorem 2.3) that if \(A \neq M_n({\mathbb C})\) is a simple AH algebra and \(A \neq {\mathcal K}\), an element \(\alpha \in KK({\mathbb C}(X),A)\) can be presented as \(\alpha=KK(\Phi)\) for some homomorphism \(\Phi:{\mathbb C} \to A\) if and only if \(\alpha\) belongs to the class \(KK_D({\mathbb C}(X),A)\), i.e. admits two properties: (1) The image \(\alpha([1])\) of the unit \(1\in {\mathbb C}(X)\), as an element in \(K_0(A)\) can be presented by an projection \(p=p^*=p^2\). (2) Each component \(\alpha_i \in KK({\mathbb C}(X_i),A)\) in the decomposition \(\alpha = \alpha_1+ \dots + \alpha_n\) either is \(0\) or maps \(K_0({\mathbb C}(X_i))^+ \setminus (0)\) into \(K_0(A)^+\setminus (0)\), the strictly positive property. As the main consequence, the author proves a characterization theorem (Theorem 3.3): If the algebra \(A\) admits a unique trace \(\tau\), then for every element \(\alpha\) with the two properties in the main result above, and for any strictly positive probability measure \(\lambda\in\mathcal (X)\), there is an (injective) homomorphism \(\Phi : {\mathbb C}(X) \to A\) such that: \(KK(\Phi) = \alpha\) and \(m_\Phi := \tau\circ \Phi = \lambda\).
scientific article; zbMATH DE number 1384365
scientific article