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On the classification of nuclear \(\text{C}^*\)-algebras. (English) Zbl 1031.46070

There is a fruitful interaction between KK-theory and the classification program, initialized by L. G. Brown, R. G. Douglas and P. A. Fillmore [Ann. of Math. 105, 265-324 (1977; Zbl 0376.46036)] and G. A. Elliot [J. Algebra 38, 29-44 (1976; Zbl 0323.46063)]. In the paper under review, the authors take this interaction even further by combining the new look of J. Cuntz KK-theory and quasidiagonality for representations.
They achieve a general uniqueness theorem: For a separable source C*-algebra \(A\) and a unital target C*-algebra \(B\) such that the pair \((A,B)\) allows nuclearly absorbing and quasidiagonal unital representation \(\gamma : A \to M(\mathcal K(H) \otimes B)\) (Theorem 2.22). Under that condition, the authors prove in Theorem 4.5 that if \(\varphi, \psi : A \to B \) are two nuclear *-homomorphisms inducing the same elements in the nuclear KK-theory \(KK_{\text{nuclear}}(A,B)\) then \(\varphi\) is stably approximately unitarily equivalent to \(\psi\) up to adding “finite pieces” of \(\gamma\). This uniqueness theorem does not depend on the Universal Coefficient Theorem (UCT) [J. Rosenberg and C. Schochet, Duke J. Math. 55, 431-474 (1987; Zbl 0644.46051)], nor on nuclearity of the involved C*-algebras.
In their existence Theorem 5.5, elements of \(KK_{\text{nuclear}}(A,B)\) are realized as a difference of nuclear completely positive contractive maps from \(A\) to \(M_N(A)\).
The main application of the uniqueness and existence theorems is Theorem 6.12 stating that up to an isomorphism, there is only one unital, separable nuclear and simple tracial \(AF\) C*-algebra satisfying UCT with \(K_0(A) \cong \mathbb Q\) and \(K_1(A) = G\), for a countable fixed arbitrary group \(G\), and therefore every such C*-algebra falls in the well-studied class of \(AD\)-algebras of real rank 0.

MSC:

46L35 Classifications of \(C^*\)-algebras
19K14 \(K_0\) as an ordered group, traces
46L80 \(K\)-theory and operator algebras (including cyclic theory)
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