Journal of Operator Theory

Volume 46, Issue 2, Fall 2001 pp. 337-353.

The K-groups of

$C(M)\times_\theta\bbb Z_p$ for certain pairs

$(M,\theta)$

Authors: Yifeng Xue
Author institution: Department of Mathematics, East China University of Science and Technology, Shanghai 200237, P.R. China

Summary: Let

$M$ be a connected, compact metric space with

$\dim M\le 2p-1$

$$p\ge 2$ is a prime$ and let

$\theta$ be a homeomorphism of

$M$ to itself with period~

$p$ . Suppose that

$\dim M_\theta\le 2$ ,

$H^2(M_\theta,\bbb Z)$

$\cong 0$ and that

$\bigoplus\limits^{2p-1}_{j=0}H^j(M/\theta,\bbb Z)$ is finitely generated and torsion-free;

$H^0(M_\theta,\bbb Z)$ is finitely generated. If

$\theta$ is regular and

$H^{2j+1}(M/\theta,\bbb Z)\cong 0$ ,

$1\le j\le p-1$ or

$\theta$ is strongly regular and

$M_\theta$ is connected, then

$\eqalign{ {\rm K}_0(C(M)\times_\theta\bbb Z_p)& \cong {\rm K}^0(M/\theta)\oplus\bigoplus\limits^{p-2}_{j=0}H^0(M_\theta,\bbb Z)\cr {\rm K}_1(C(M)\times_\theta\bbb Z_p)& \cong {\rm K}^{-1}(M/\theta)\oplus\bigoplus\limits^{p-2}_{j=0}H^1 (M_\theta,\bbb Z). \cr}$ The result leads us to compute some interesting examples when

$M$ is a sphere or a torus.

Keywords:

$K$ -groups of

$C^*$ -algebras, crossed product of

$C^*$ -algebras, stable rank, Cech cohomology groups, regular self-homeomorphism of prime period

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