Suppose that A is a separable C*-algebra and that G∗ is a (graded) subgroup of the ℤ/2-graded group K∗(A). Then there is a natural short exact sequence
0 → G∗ → K∗(A) → K∗(A)/G∗ → 0.
In this note we demonstrate how to geometrically realize this sequence at the level of C*-algebras. As a result, we KK-theoretically decompose A as
0 → A ⊗ [cursive]K → Aƒ → SAt → 0
where K∗(At) is the torsion subgroup of K∗(A) and K∗(Aƒ) is its torsionfree quotient. Then we further decompose At: it is KK-equivalent to ⊕pAp where K∗(Ap) is the p-primary subgroup of the torsion subgroup of K∗(A). We then apply this realization to study the Kasparov group K*(A) and related objects.
Algebra | Algebraic Geometry
C. Schochet, Geometric realization and K-theoretic decomposition of C*-algebras, International Journal of Mathematics 12(3) (2001), 373-381.