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28 R. MeyerA and B – at least for many A and B. This is the point of the Universal CoefficientTheorem.The Kasparov product provides a canonical homomorphism of graded groups W KK .A; B/ ! Hom K .A/; K .B/ ;where Hom denotes the Z=2-graded Abelian group of all group homomorphismsK .A/ ! K .B/. There are topological reasons why cannot always be invertible:since Hom is not exact, the bifunctor Hom K .A/; K .B/ would not be exact oncp-split extensions. A construction of Lawrence Brown provides another natural map W ker ! Ext K C1 .A/; K .B/ :The following theorem is due to Jonathan Rosenberg and Claude Schochet [53], [51];see also [5].Theorem 57. The following are equivalent for a separable C -algebra A:(a) KK .A; B/ D 0 for all B 22 KK with K .B/ D 0;(b) the map is surjective and is bijective for all B 22 KK;(c) for all B 22 KK, there is a short exact sequence of Z=2-graded Abelian groupsExt K C1 .A/; K .B/ KK .A; B/ Hom K .A/; K .B/ :(d) A belongs to the smallest class of C -algebras that contains C and is closed underKK-equivalence, suspensions, countable direct sums, and cp-split extensions;(e) A is KK-equivalent to C 0 .X/ for some pointed compact metrisable space X.If these conditions are satisfied, then the extension in (c) is natural and splits, but thesection is not natural.The class of C -algebras with these properties is also called the bootstrap classbecause of description (d). Alternatively, we may say that they satisfy the UniversalCoefficient Theorem because of (c). Since commutative C -algebras are nuclear, (e)implies that the natural map A ˝max B ! A ˝min B is a KK-equivalence if A or Bbelongs to the bootstrap class [55]. This fails for some A, so that the Universal CoefficientTheorem does not hold for all A. Remarkably, this is the only obstruction to theUniversal Coefficient Theorem known at the moment: we know no nuclear C -algebrathat does not satisfy the Universal Coefficient Theorem. As a result, we can expressKK .A; B/ using only K .A/ and K .B/ for many A and B.When we restrict attention to nuclear C -algebras, then the bootstrap class is closedunder various operations like tensor products, arbitrary extensions and inductive limits(without requiring any cp-sections), and under crossed products by torsion-freeamenable groups. Remarkably, there are no general results about crossed products byfinite groups.