#### Document Type

Article

#### Abstract

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* → *A*_{ƒ} → *SA*_{t} → 0

where *K*∗(*A*_{t}) is the torsion subgroup of *K*∗(*A*) and *K*∗(*A*_{ƒ}) is its torsionfree quotient. Then we further decompose *A*_{t}: it is *KK*-equivalent to ⊕_{p}*A*_{p} where *K*∗(*A*_{p}) 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.

#### Disciplines

Algebra | Algebraic Geometry

#### Recommended Citation

C. Schochet, Geometric realization and *K*-theoretic decomposition of *C**-algebras, *International Journal of Mathematics* **12(3)** (2001), 373-381.

## Comments

This is the final accepted manuscript copy, derived from arXiv.org (http://arxiv.org/abs/math/0107042v1), of an electronic version of an article published as Geometric realization and

K-theoretic decomposition ofC*-algebras,International Journal of Mathematics12(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.