Quasidiagonal Extensions and Sequentially Trivial Asymptotic Homomorphisms

The Connes–Higson construction is an isomorphism

“…An argument almost identical with the one used to prove Lemma 5.3 of [MT1] shows that ½q B#K 3 e j j is also independent of the chosen discretization of # j: Once this is established it is clear that a homotopy of asymptotic homomorphisms A-QðBÞ G #K gives rise, by an appropriate choice of unit sequence, to a homotopy which shows that ½q B#K 3 e j jAExtðA; B#KÞ h only depends on the homotopy class of j: & It follows that we have the desired map a : ½½A; QðBÞ G #K-ExtðA; B#KÞ h which is easily seen to be a semi-group homomorphism.…”

Discrete asymptotic homomorphisms in E-theory and KK-theory

“…This was pointed out in [12], but is really something which follows from [11], [7] and [9]. Let us assume that A is in the Bootstrap-category for which the UCT holds, cf.…”

“…[13], [6]. Then Pext(K * (A), K * −1 (B)) can be realized as a subgroup of Ext −1 (SA, B), and it was shown in [12] that the isomorphism CH takes this subgroup onto the range of It follows from this, the exactness of the diagram(s) in Theorem 6.1 and the version of the UCT from [6], that the range of the map d •B in Theorem 6.1 can be identified with KL (A, B). In other words, by unsplicing the first six-term exact sequence of Theorem 6.1 we obtain an extension of groups which is the same as the extension in the UCT theorem, in the form it was given in [6].…”

$E$-Theory is a Special Case of $KK$-Theory

“…Recall (see [10]) that a discretization, f' t n g 1 n¼1 , of an asymptotic homomorphism ' ¼ ð' t Þ t2½1;1Þ : A ! B is given by a sequence t 1 < t 2 < t 3 < .…”

“…in ½1; 1Þ such that (d1) lim n!1 t n ¼ 1, and (d2) lim n!1 sup t2½t n ;t nþ1 k' t ðaÞ À ' t n ðaÞk ¼ 0 for all a 2 A. When A is separable, discretizations always exist; see [10].…”