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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]KAƒSAt → 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


This is the final accepted manuscript copy, derived from (, of an electronic version of an article published as Geometric realization and K-theoretic decomposition of C*-algebras, International Journal of Mathematics 12(3) (2001), 373-381 [DOI: 10.1142/S0129167X01000794], © Copyright World Scientific Publishing Company, International Journal of Mathematics.

1991 Mathematics Subject Classification: Primary 46L80, 19K35, 46L85.