By Klaus Thomsen
Quantity 206, quantity 970 (end of volume).
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Extra resources for C*-Algebras of Homoclinic and Heteroclinic Structure in Expensive Dynamics
Of continuous surjective group endomorphisms such that SQ is isomorphic, as a topological group, to the projective limit of the sequence Tn ϕ1 Tn ϕ2 Tn ϕ3 Tn ϕ4 . . In particular, SQ is connected and divisible. Proof. For each k, equip the subgroup Ωk = Qt −k (Zn ) + Qt −k+1 (Zn ) + · · · + Qt k−1 (Zn ) + Qt k (Zn ) of Qn with the discrete topology. The dual group SQ of SQ can then be identiﬁed with the union ∞ k=1 Ωk , cf. 2 of [KS]. It follows that SQ is isomorphic to the corresponding projective limit Ω1 Since Ωk Ω2 Zn , and hence Ωk Ω3 Ω4 ...
30. Let Q ∈ Mn (Z)∩Gln (Q) be hyperbolic. 19) Zn Q Zn Q Zn Q ... , 40 3. THE HOMOCLINIC ALGEBRA OF EXPANSIVE ACTIONS where Qt is the transpose of Q. It can happen that these inductive limit groups are not isomorphic. This is for example the case when Q= 65 7 , 24 67 cf. 6 of [BJKR]. In this case the homoclinic group of σQ is not isomorphic to SQ (as it is in the case where Q ∈ Gln (Z)). e. as dimension groups) the arguments for the stronger statement have been included in Appendix C. Presumably the phenomenon is not exceptional at all; the methods explored in Appendix C can be used to ﬁnd other examples, albeit in a rather unsystematic way.
9 implies that (π• , Eπ , 0) is a Kasparov Af (X, E) − Af (X , E ) module, and hence the triple deﬁnes an element [π] ∈ KK (Af (X, E), Af (X , E )) . 10. Assume that conditions 1 and 4 hold, and that E is second countable. Then the element [π] ∈ KK (Af (X, E), Af (X , E )) is represented by a ∗homomorphism Af (X, E) → Af (X , E ) ⊗ K. Proof. To simplify notation, set A = Af (X, E) and B = Af (X , E ). Using the notation from [K-JT] we have that [π] is represented by (π• , Eπ , 0) ⊕ (0, HB , 0).
C*-Algebras of Homoclinic and Heteroclinic Structure in Expensive Dynamics by Klaus Thomsen